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{userdict/md get type/dicttype eq{userdict begin md length 10 add md
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atan/pa X newpath clippath mark{transform{itransform moveto}}{transform{
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transform{itransform 6 2 roll itransform 6 2 roll itransform 6 2 roll
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pop ct 39 0 put 10 fz 0 fs 2 F/|______Courier fnt invertflag{PaintBlack}
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get ppr 1 get neg sub neg ppr 2 get ppr 0 get neg sub neg TR}if xflip
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neg 0 TR}if yflip xflip not and{ppr 1 get neg ppr 0 get neg TR}if}{
noflips{TR pop pop 270 rotate 1 -1 scale}if xflip yflip and{TR pop pop
90 rotate 1 -1 scale ppr 3 get ppr 1 get neg sub neg ppr 2 get ppr 0 get
neg sub neg TR}if xflip yflip not and{TR pop pop 90 rotate ppr 3 get ppr
1 get neg sub neg 0 TR}if yflip xflip not and{TR pop pop 270 rotate ppr
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rwiSeen{rwi urx llx sub div rhiSeen{rhi ury lly sub div}{dup}ifelse
scale llx neg lly neg TR}{rhiSeen{rhi ury lly sub div dup scale llx neg
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ury lineto llx ury lineto closepath clip}if/showpage{}N/erasepage{}N
/setpagedevice{pop}N/copypage{}N newpath}N/@endspecial{count ocount sub{
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matrix currentmatrix N TR xrad yrad scale 0 0 1 startangle endangle arc
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%%EndProcSet
%%BeginFont: CMBX10
%!PS-AdobeFont-1.1: CMBX10 1.00B
%%CreationDate: 1992 Feb 19 19:54:06
% Copyright (C) 1997 American Mathematical Society. All Rights Reserved.
11 dict begin
/FontInfo 7 dict dup begin
/version (1.00B) readonly def
/Notice (Copyright (C) 1997 American Mathematical Society. All Rights Reserved) readonly def
/FullName (CMBX10) readonly def
/FamilyName (Computer Modern) readonly def
/Weight (Bold) readonly def
/ItalicAngle 0 def
/isFixedPitch false def
end readonly def
/FontName /CMBX10 def
/PaintType 0 def
/FontType 1 def
/FontMatrix [0.001 0 0 0.001 0 0] readonly def
/Encoding 256 array
0 1 255 {1 index exch /.notdef put} for
dup 0 /.notdef put
readonly def
/FontBBox{-301 -250 1164 946}readonly def
/UniqueID 5000768 def
currentdict end
currentfile eexec
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cleartomark
%%EndFont 
%%BeginFont: CMCSC10
%!PS-AdobeFont-1.1: CMCSC10 1.0
%%CreationDate: 1991 Aug 18 17:46:49
% Copyright (C) 1997 American Mathematical Society. All Rights Reserved.
11 dict begin
/FontInfo 7 dict dup begin
/version (1.0) readonly def
/Notice (Copyright (C) 1997 American Mathematical Society. All Rights Reserved) readonly def
/FullName (CMCSC10) readonly def
/FamilyName (Computer Modern) readonly def
/Weight (Medium) readonly def
/ItalicAngle 0 def
/isFixedPitch false def
end readonly def
/FontName /CMCSC10 def
/PaintType 0 def
/FontType 1 def
/FontMatrix [0.001 0 0 0.001 0 0] readonly def
/Encoding 256 array
0 1 255 {1 index exch /.notdef put} for
dup 0 /.notdef put
readonly def
/FontBBox{14 -250 1077 750}readonly def
/UniqueID 5000772 def
currentdict end
currentfile eexec
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cleartomark
%%EndFont 
%%BeginFont: CMSL12
%!PS-AdobeFont-1.1: CMSL12 1.0
%%CreationDate: 1991 Aug 20 16:40:41
% Copyright (C) 1997 American Mathematical Society. All Rights Reserved.
11 dict begin
/FontInfo 7 dict dup begin
/version (1.0) readonly def
/Notice (Copyright (C) 1997 American Mathematical Society. All Rights Reserved) readonly def
/FullName (CMSL12) readonly def
/FamilyName (Computer Modern) readonly def
/Weight (Medium) readonly def
/ItalicAngle -9.46 def
/isFixedPitch false def
end readonly def
/FontName /CMSL12 def
/PaintType 0 def
/FontType 1 def
/FontMatrix [0.001 0 0 0.001 0 0] readonly def
/Encoding 256 array
0 1 255 {1 index exch /.notdef put} for
dup 0 /.notdef put
readonly def
/FontBBox{-56 -251 1102 750}readonly def
/UniqueID 5000799 def
currentdict end
currentfile eexec
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cleartomark
%%EndFont 
%%BeginFont: MSAM10
%!PS-AdobeFont-1.1: MSAM10 2.1
%%CreationDate: 1993 Sep 17 09:05:00
% Math Symbol fonts were designed by the American Mathematical Society.
% Copyright (C) 1997 American Mathematical Society. All Rights Reserved.
11 dict begin
/FontInfo 7 dict dup begin
/version (2.1) readonly def
/Notice (Copyright (C) 1997 American Mathematical Society. All Rights Reserved) readonly def
/FullName (MSAM10) readonly def
/FamilyName (Euler) readonly def
/Weight (Medium) readonly def
/ItalicAngle 0 def
/isFixedPitch false def
end readonly def
/FontName /MSAM10 def
/PaintType 0 def
/FontType 1 def
/FontMatrix [0.001 0 0 0.001 0 0] readonly def
/Encoding 256 array
0 1 255 {1 index exch /.notdef put} for
dup 3 /square put
readonly def
/FontBBox{8 -463 1331 1003}readonly def
/UniqueID 5031981 def
currentdict end
currentfile eexec
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cleartomark
%%EndFont 
%%BeginFont: CMMI5
%!PS-AdobeFont-1.1: CMMI5 1.100
%%CreationDate: 1996 Aug 02 08:21:10
% Copyright (C) 1997 American Mathematical Society. All Rights Reserved.
11 dict begin
/FontInfo 7 dict dup begin
/version (1.100) readonly def
/Notice (Copyright (C) 1997 American Mathematical Society. All Rights Reserved) readonly def
/FullName (CMMI5) readonly def
/FamilyName (Computer Modern) readonly def
/Weight (Medium) readonly def
/ItalicAngle -14.04 def
/isFixedPitch false def
end readonly def
/FontName /CMMI5 def
/PaintType 0 def
/FontType 1 def
/FontMatrix [0.001 0 0 0.001 0 0] readonly def
/Encoding 256 array
0 1 255 {1 index exch /.notdef put} for
dup 0 /.notdef put
readonly def
/FontBBox{37 -250 1349 750}readonly def
/UniqueID 5087380 def
currentdict end
currentfile eexec
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cleartomark
%%EndFont 
%%BeginFont: MSBM7
%!PS-AdobeFont-1.1: MSBM7 2.1
%%CreationDate: 1992 Oct 17 08:30:50
% Math Symbol fonts were designed by the American Mathematical Society.
% Copyright (C) 1997 American Mathematical Society. All Rights Reserved.
11 dict begin
/FontInfo 7 dict dup begin
/version (2.1) readonly def
/Notice (Copyright (C) 1997 American Mathematical Society. All Rights Reserved) readonly def
/FullName (MSBM7) readonly def
/FamilyName (Euler) readonly def
/Weight (Medium) readonly def
/ItalicAngle 0 def
/isFixedPitch false def
end readonly def
/FontName /MSBM7 def
/PaintType 0 def
/FontType 1 def
/FontMatrix [0.001 0 0 0.001 0 0] readonly def
/Encoding 256 array
0 1 255 {1 index exch /.notdef put} for
dup 84 /T put
dup 90 /Z put
readonly def
/FontBBox{0 -504 2615 1004}readonly def
/UniqueID 5032014 def
currentdict end
currentfile eexec
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cleartomark
%%EndFont 
%%BeginFont: CMEX10
%!PS-AdobeFont-1.1: CMEX10 1.00
%%CreationDate: 1992 Jul 23 21:22:48
% Copyright (C) 1997 American Mathematical Society. All Rights Reserved.
11 dict begin
/FontInfo 7 dict dup begin
/version (1.00) readonly def
/Notice (Copyright (C) 1997 American Mathematical Society. All Rights Reserved) readonly def
/FullName (CMEX10) readonly def
/FamilyName (Computer Modern) readonly def
/Weight (Medium) readonly def
/ItalicAngle 0 def
/isFixedPitch false def
end readonly def
/FontName /CMEX10 def
/PaintType 0 def
/FontType 1 def
/FontMatrix [0.001 0 0 0.001 0 0] readonly def
/Encoding 256 array
0 1 255 {1 index exch /.notdef put} for
dup 12 /vextendsingle put
dup 16 /parenleftBig put
dup 17 /parenrightBig put
dup 26 /braceleftbigg put
dup 56 /bracelefttp put
dup 57 /bracerighttp put
dup 58 /braceleftbt put
dup 59 /bracerightbt put
dup 60 /braceleftmid put
dup 61 /bracerightmid put
dup 62 /braceex put
dup 68 /angbracketleftBig put
dup 69 /angbracketrightBig put
dup 80 /summationtext put
dup 82 /integraltext put
dup 88 /summationdisplay put
dup 90 /integraldisplay put
dup 101 /tildewide put
dup 104 /bracketleftBig put
dup 105 /bracketrightBig put
dup 110 /braceleftBig put
dup 111 /bracerightBig put
dup 112 /radicalbig put
readonly def
/FontBBox{-24 -2960 1454 772}readonly def
/UniqueID 5000774 def
currentdict end
currentfile eexec
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cleartomark
%%EndFont 
%%BeginFont: CMMI7
%!PS-AdobeFont-1.1: CMMI7 1.100
%%CreationDate: 1996 Jul 23 07:53:53
% Copyright (C) 1997 American Mathematical Society. All Rights Reserved.
11 dict begin
/FontInfo 7 dict dup begin
/version (1.100) readonly def
/Notice (Copyright (C) 1997 American Mathematical Society. All Rights Reserved) readonly def
/FullName (CMMI7) readonly def
/FamilyName (Computer Modern) readonly def
/Weight (Medium) readonly def
/ItalicAngle -14.04 def
/isFixedPitch false def
end readonly def
/FontName /CMMI7 def
/PaintType 0 def
/FontType 1 def
/FontMatrix [0.001 0 0 0.001 0 0] readonly def
/Encoding 256 array
0 1 255 {1 index exch /.notdef put} for
dup 0 /.notdef put
readonly def
/FontBBox{0 -250 1171 750}readonly def
/UniqueID 5087382 def
currentdict end
currentfile eexec
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cleartomark
%%EndFont 
%%BeginFont: CMSY7
%!PS-AdobeFont-1.1: CMSY7 1.0
%%CreationDate: 1991 Aug 15 07:21:52
% Copyright (C) 1997 American Mathematical Society. All Rights Reserved.
11 dict begin
/FontInfo 7 dict dup begin
/version (1.0) readonly def
/Notice (Copyright (C) 1997 American Mathematical Society. All Rights Reserved) readonly def
/FullName (CMSY7) readonly def
/FamilyName (Computer Modern) readonly def
/Weight (Medium) readonly def
/ItalicAngle -14.035 def
/isFixedPitch false def
end readonly def
/FontName /CMSY7 def
/PaintType 0 def
/FontType 1 def
/FontMatrix [0.001 0 0 0.001 0 0] readonly def
/Encoding 256 array
0 1 255 {1 index exch /.notdef put} for
dup 0 /.notdef put
readonly def
/FontBBox{-15 -951 1252 782}readonly def
/UniqueID 5000817 def
currentdict end
currentfile eexec
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cleartomark
%%EndFont 
%%BeginFont: MSBM10
%!PS-AdobeFont-1.1: MSBM10 2.1
%%CreationDate: 1993 Sep 17 11:10:37
% Math Symbol fonts were designed by the American Mathematical Society.
% Copyright (C) 1997 American Mathematical Society. All Rights Reserved.
11 dict begin
/FontInfo 7 dict dup begin
/version (2.1) readonly def
/Notice (Copyright (C) 1997 American Mathematical Society. All Rights Reserved) readonly def
/FullName (MSBM10) readonly def
/FamilyName (Euler) readonly def
/Weight (Medium) readonly def
/ItalicAngle 0 def
/isFixedPitch false def
end readonly def
/FontName /MSBM10 def
/PaintType 0 def
/FontType 1 def
/FontMatrix [0.001 0 0 0.001 0 0] readonly def
/Encoding 256 array
0 1 255 {1 index exch /.notdef put} for
dup 67 /C put
dup 78 /N put
dup 81 /Q put
dup 82 /R put
dup 84 /T put
dup 90 /Z put
readonly def
/FontBBox{-55 -420 2343 920}readonly def
/UniqueID 5031982 def
currentdict end
currentfile eexec
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cleartomark
%%EndFont 
%%BeginFont: CMSY10
%!PS-AdobeFont-1.1: CMSY10 1.0
%%CreationDate: 1991 Aug 15 07:20:57
% Copyright (C) 1997 American Mathematical Society. All Rights Reserved.
11 dict begin
/FontInfo 7 dict dup begin
/version (1.0) readonly def
/Notice (Copyright (C) 1997 American Mathematical Society. All Rights Reserved) readonly def
/FullName (CMSY10) readonly def
/FamilyName (Computer Modern) readonly def
/Weight (Medium) readonly def
/ItalicAngle -14.035 def
/isFixedPitch false def
end readonly def
/FontName /CMSY10 def
/PaintType 0 def
/FontType 1 def
/FontMatrix [0.001 0 0 0.001 0 0] readonly def
/Encoding 256 array
0 1 255 {1 index exch /.notdef put} for
dup 0 /.notdef put
readonly def
/FontBBox{-29 -960 1116 775}readonly def
/UniqueID 5000820 def
currentdict end
currentfile eexec
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cleartomark
%%EndFont 
%%BeginFont: CMMI10
%!PS-AdobeFont-1.1: CMMI10 1.100
%%CreationDate: 1996 Jul 23 07:53:57
% Copyright (C) 1997 American Mathematical Society. All Rights Reserved.
11 dict begin
/FontInfo 7 dict dup begin
/version (1.100) readonly def
/Notice (Copyright (C) 1997 American Mathematical Society. All Rights Reserved) readonly def
/FullName (CMMI10) readonly def
/FamilyName (Computer Modern) readonly def
/Weight (Medium) readonly def
/ItalicAngle -14.04 def
/isFixedPitch false def
end readonly def
/FontName /CMMI10 def
/PaintType 0 def
/FontType 1 def
/FontMatrix [0.001 0 0 0.001 0 0] readonly def
/Encoding 256 array
0 1 255 {1 index exch /.notdef put} for
dup 0 /.notdef put
readonly def
/FontBBox{-32 -250 1048 750}readonly def
/UniqueID 5087385 def
currentdict end
currentfile eexec
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cleartomark
%%EndFont 
%%BeginFont: CMSY6
%!PS-AdobeFont-1.1: CMSY6 1.0
%%CreationDate: 1991 Aug 15 07:21:34
% Copyright (C) 1997 American Mathematical Society. All Rights Reserved.
11 dict begin
/FontInfo 7 dict dup begin
/version (1.0) readonly def
/Notice (Copyright (C) 1997 American Mathematical Society. All Rights Reserved) readonly def
/FullName (CMSY6) readonly def
/FamilyName (Computer Modern) readonly def
/Weight (Medium) readonly def
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end readonly def
/FontName /CMSY6 def
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/FontMatrix [0.001 0 0 0.001 0 0] readonly def
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0 1 255 {1 index exch /.notdef put} for
dup 0 /.notdef put
readonly def
/FontBBox{-4 -948 1329 786}readonly def
/UniqueID 5000816 def
currentdict end
currentfile eexec
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cleartomark
%%EndFont 
%%BeginFont: CMMI6
%!PS-AdobeFont-1.1: CMMI6 1.100
%%CreationDate: 1996 Jul 23 07:53:52
% Copyright (C) 1997 American Mathematical Society. All Rights Reserved.
11 dict begin
/FontInfo 7 dict dup begin
/version (1.100) readonly def
/Notice (Copyright (C) 1997 American Mathematical Society. All Rights Reserved) readonly def
/FullName (CMMI6) readonly def
/FamilyName (Computer Modern) readonly def
/Weight (Medium) readonly def
/ItalicAngle -14.04 def
/isFixedPitch false def
end readonly def
/FontName /CMMI6 def
/PaintType 0 def
/FontType 1 def
/FontMatrix [0.001 0 0 0.001 0 0] readonly def
/Encoding 256 array
0 1 255 {1 index exch /.notdef put} for
dup 0 /.notdef put
readonly def
/FontBBox{11 -250 1241 750}readonly def
/UniqueID 5087381 def
currentdict end
currentfile eexec
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cleartomark
%%EndFont 
%%BeginFont: CMTI12
%!PS-AdobeFont-1.1: CMTI12 1.0
%%CreationDate: 1991 Aug 18 21:06:53
% Copyright (C) 1997 American Mathematical Society. All Rights Reserved.
11 dict begin
/FontInfo 7 dict dup begin
/version (1.0) readonly def
/Notice (Copyright (C) 1997 American Mathematical Society. All Rights Reserved) readonly def
/FullName (CMTI12) readonly def
/FamilyName (Computer Modern) readonly def
/Weight (Medium) readonly def
/ItalicAngle -14.04 def
/isFixedPitch false def
end readonly def
/FontName /CMTI12 def
/PaintType 0 def
/FontType 1 def
/FontMatrix [0.001 0 0 0.001 0 0] readonly def
/Encoding 256 array
0 1 255 {1 index exch /.notdef put} for
dup 0 /.notdef put
readonly def
/FontBBox{-36 -251 1103 750}readonly def
/UniqueID 5000829 def
currentdict end
currentfile eexec
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cleartomark
%%EndFont 
%%BeginFont: CMR6
%!PS-AdobeFont-1.1: CMR6 1.0
%%CreationDate: 1991 Aug 20 16:39:02
% Copyright (C) 1997 American Mathematical Society. All Rights Reserved.
11 dict begin
/FontInfo 7 dict dup begin
/version (1.0) readonly def
/Notice (Copyright (C) 1997 American Mathematical Society. All Rights Reserved) readonly def
/FullName (CMR6) readonly def
/FamilyName (Computer Modern) readonly def
/Weight (Medium) readonly def
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end readonly def
/FontName /CMR6 def
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/FontMatrix [0.001 0 0 0.001 0 0] readonly def
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0 1 255 {1 index exch /.notdef put} for
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readonly def
/FontBBox{-20 -250 1193 750}readonly def
/UniqueID 5000789 def
currentdict end
currentfile eexec
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cleartomark
%%EndFont 
%%BeginFont: CMR9
%!PS-AdobeFont-1.1: CMR9 1.0
%%CreationDate: 1991 Aug 20 16:39:59
% Copyright (C) 1997 American Mathematical Society. All Rights Reserved.
11 dict begin
/FontInfo 7 dict dup begin
/version (1.0) readonly def
/Notice (Copyright (C) 1997 American Mathematical Society. All Rights Reserved) readonly def
/FullName (CMR9) readonly def
/FamilyName (Computer Modern) readonly def
/Weight (Medium) readonly def
/ItalicAngle 0 def
/isFixedPitch false def
end readonly def
/FontName /CMR9 def
/PaintType 0 def
/FontType 1 def
/FontMatrix [0.001 0 0 0.001 0 0] readonly def
/Encoding 256 array
0 1 255 {1 index exch /.notdef put} for
dup 0 /.notdef put
readonly def
/FontBBox{-39 -250 1036 750}readonly def
/UniqueID 5000792 def
currentdict end
currentfile eexec
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cleartomark
%%EndFont 
%%BeginFont: CMTT10
%!PS-AdobeFont-1.1: CMTT10 1.00B
%%CreationDate: 1992 Apr 26 10:42:42
% Copyright (C) 1997 American Mathematical Society. All Rights Reserved.
11 dict begin
/FontInfo 7 dict dup begin
/version (1.00B) readonly def
/Notice (Copyright (C) 1997 American Mathematical Society. All Rights Reserved) readonly def
/FullName (CMTT10) readonly def
/FamilyName (Computer Modern) readonly def
/Weight (Medium) readonly def
/ItalicAngle 0 def
/isFixedPitch true def
end readonly def
/FontName /CMTT10 def
/PaintType 0 def
/FontType 1 def
/FontMatrix [0.001 0 0 0.001 0 0] readonly def
/Encoding 256 array
0 1 255 {1 index exch /.notdef put} for
dup 0 /.notdef put
readonly def
/FontBBox{-4 -235 731 800}readonly def
/UniqueID 5000832 def
currentdict end
currentfile eexec
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%%EndFont 
%%BeginFont: CMSY8
%!PS-AdobeFont-1.1: CMSY8 1.0
%%CreationDate: 1991 Aug 15 07:22:10
% Copyright (C) 1997 American Mathematical Society. All Rights Reserved.
11 dict begin
/FontInfo 7 dict dup begin
/version (1.0) readonly def
/Notice (Copyright (C) 1997 American Mathematical Society. All Rights Reserved) readonly def
/FullName (CMSY8) readonly def
/FamilyName (Computer Modern) readonly def
/Weight (Medium) readonly def
/ItalicAngle -14.035 def
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/FontName /CMSY8 def
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/FontType 1 def
/FontMatrix [0.001 0 0 0.001 0 0] readonly def
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/FontBBox{-30 -955 1185 779}readonly def
/UniqueID 5000818 def
currentdict end
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cleartomark
%%EndFont 
%%BeginFont: CMR8
%!PS-AdobeFont-1.1: CMR8 1.0
%%CreationDate: 1991 Aug 20 16:39:40
% Copyright (C) 1997 American Mathematical Society. All Rights Reserved.
11 dict begin
/FontInfo 7 dict dup begin
/version (1.0) readonly def
/Notice (Copyright (C) 1997 American Mathematical Society. All Rights Reserved) readonly def
/FullName (CMR8) readonly def
/FamilyName (Computer Modern) readonly def
/Weight (Medium) readonly def
/ItalicAngle 0 def
/isFixedPitch false def
end readonly def
/FontName /CMR8 def
/PaintType 0 def
/FontType 1 def
/FontMatrix [0.001 0 0 0.001 0 0] readonly def
/Encoding 256 array
0 1 255 {1 index exch /.notdef put} for
dup 0 /.notdef put
readonly def
/FontBBox{-36 -250 1070 750}readonly def
/UniqueID 5000791 def
currentdict end
currentfile eexec
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cleartomark
%%EndFont 
%%BeginFont: CMMI8
%!PS-AdobeFont-1.1: CMMI8 1.100
%%CreationDate: 1996 Jul 23 07:53:54
% Copyright (C) 1997 American Mathematical Society. All Rights Reserved.
11 dict begin
/FontInfo 7 dict dup begin
/version (1.100) readonly def
/Notice (Copyright (C) 1997 American Mathematical Society. All Rights Reserved) readonly def
/FullName (CMMI8) readonly def
/FamilyName (Computer Modern) readonly def
/Weight (Medium) readonly def
/ItalicAngle -14.04 def
/isFixedPitch false def
end readonly def
/FontName /CMMI8 def
/PaintType 0 def
/FontType 1 def
/FontMatrix [0.001 0 0 0.001 0 0] readonly def
/Encoding 256 array
0 1 255 {1 index exch /.notdef put} for
dup 0 /.notdef put
readonly def
/FontBBox{-24 -250 1110 750}readonly def
/UniqueID 5087383 def
currentdict end
currentfile eexec
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cleartomark
%%EndFont 
%%BeginFont: CMMI12
%!PS-AdobeFont-1.1: CMMI12 1.100
%%CreationDate: 1996 Jul 27 08:57:55
% Copyright (C) 1997 American Mathematical Society. All Rights Reserved.
11 dict begin
/FontInfo 7 dict dup begin
/version (1.100) readonly def
/Notice (Copyright (C) 1997 American Mathematical Society. All Rights Reserved) readonly def
/FullName (CMMI12) readonly def
/FamilyName (Computer Modern) readonly def
/Weight (Medium) readonly def
/ItalicAngle -14.04 def
/isFixedPitch false def
end readonly def
/FontName /CMMI12 def
/PaintType 0 def
/FontType 1 def
/FontMatrix [0.001 0 0 0.001 0 0] readonly def
/Encoding 256 array
0 1 255 {1 index exch /.notdef put} for
dup 0 /.notdef put
readonly def
/FontBBox{-30 -250 1026 750}readonly def
/UniqueID 5087386 def
currentdict end
currentfile eexec
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cleartomark
%%EndFont 
%%BeginFont: CMR12
%!PS-AdobeFont-1.1: CMR12 1.0
%%CreationDate: 1991 Aug 20 16:38:05
% Copyright (C) 1997 American Mathematical Society. All Rights Reserved.
11 dict begin
/FontInfo 7 dict dup begin
/version (1.0) readonly def
/Notice (Copyright (C) 1997 American Mathematical Society. All Rights Reserved) readonly def
/FullName (CMR12) readonly def
/FamilyName (Computer Modern) readonly def
/Weight (Medium) readonly def
/ItalicAngle 0 def
/isFixedPitch false def
end readonly def
/FontName /CMR12 def
/PaintType 0 def
/FontType 1 def
/FontMatrix [0.001 0 0 0.001 0 0] readonly def
/Encoding 256 array
0 1 255 {1 index exch /.notdef put} for
dup 0 /.notdef put
readonly def
/FontBBox{-34 -251 988 750}readonly def
/UniqueID 5000794 def
currentdict end
currentfile eexec
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% Copyright (C) 1997 American Mathematical Society. All Rights Reserved.
11 dict begin
/FontInfo 7 dict dup begin
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/FullName (CMR7) readonly def
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currentdict end
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cleartomark
%%EndFont 
%%BeginFont: CMR10
%!PS-AdobeFont-1.1: CMR10 1.00B
%%CreationDate: 1992 Feb 19 19:54:52
% Copyright (C) 1997 American Mathematical Society. All Rights Reserved.
11 dict begin
/FontInfo 7 dict dup begin
/version (1.00B) readonly def
/Notice (Copyright (C) 1997 American Mathematical Society. All Rights Reserved) readonly def
/FullName (CMR10) readonly def
/FamilyName (Computer Modern) readonly def
/Weight (Medium) readonly def
/ItalicAngle 0 def
/isFixedPitch false def
end readonly def
/FontName /CMR10 def
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/FontMatrix [0.001 0 0 0.001 0 0] readonly def
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readonly def
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currentdict end
currentfile eexec
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cleartomark
%%EndFont 
%%BeginFont: CMTI10
%!PS-AdobeFont-1.1: CMTI10 1.00B
%%CreationDate: 1992 Feb 19 19:56:16
% Copyright (C) 1997 American Mathematical Society. All Rights Reserved.
11 dict begin
/FontInfo 7 dict dup begin
/version (1.00B) readonly def
/Notice (Copyright (C) 1997 American Mathematical Society. All Rights Reserved) readonly def
/FullName (CMTI10) readonly def
/FamilyName (Computer Modern) readonly def
/Weight (Medium) readonly def
/ItalicAngle -14.04 def
/isFixedPitch false def
end readonly def
/FontName /CMTI10 def
/PaintType 0 def
/FontType 1 def
/FontMatrix [0.001 0 0 0.001 0 0] readonly def
/Encoding 256 array
0 1 255 {1 index exch /.notdef put} for
dup 0 /.notdef put
readonly def
/FontBBox{-163 -250 1146 969}readonly def
/UniqueID 5000828 def
currentdict end
currentfile eexec
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cleartomark
%%EndFont 
%%BeginFont: CMBX12
%!PS-AdobeFont-1.1: CMBX12 1.0
%%CreationDate: 1991 Aug 20 16:34:54
% Copyright (C) 1997 American Mathematical Society. All Rights Reserved.
11 dict begin
/FontInfo 7 dict dup begin
/version (1.0) readonly def
/Notice (Copyright (C) 1997 American Mathematical Society. All Rights Reserved) readonly def
/FullName (CMBX12) readonly def
/FamilyName (Computer Modern) readonly def
/Weight (Bold) readonly def
/ItalicAngle 0 def
/isFixedPitch false def
end readonly def
/FontName /CMBX12 def
/PaintType 0 def
/FontType 1 def
/FontMatrix [0.001 0 0 0.001 0 0] readonly def
/Encoding 256 array
0 1 255 {1 index exch /.notdef put} for
dup 0 /.notdef put
readonly def
/FontBBox{-53 -251 1139 750}readonly def
/UniqueID 5000769 def
currentdict end
currentfile eexec
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y Fy(a)151 2064 y Fv(2)p Fx(d)p Fu(\000)p Fv(1)317 2049
y FA(\()p Fy(')419 2064 y Fv(1)458 2049 y FA(\))28 b
Fw(6)p FA(=)f(0)p Fo(,)35 b Fw(8)p Fy(')860 2064 y Fv(1)927
2049 y Fw(2)28 b Ft(T)p Fo(.)45 b(L)-5 b(et)36 b Fw(B)1396
2064 y Fx(\015)1475 2049 y Fo(b)-5 b(e)35 b(the)g(unc)-5
b(ountable)2258 2010 y Fv(3)2327 2049 y Fo(zer)g(o-me)g(asur)g(e)34
b(Cantor)h(set)665 2255 y Fw(B)730 2270 y Fx(\015)803
2255 y FA(:=)933 2144 y Fz(n)1000 2255 y Fy(")27 b Fw(2)h
FA(\()p Fw(\000)p Fy(")1328 2270 y Fv(0)1368 2255 y Fy(;)17
b(")1458 2270 y Fv(0)1496 2255 y FA(\))45 b(:)127 b Fw(j)p
Fy(l)1790 2270 y Fv(1)1851 2255 y FA(+)22 b Fy("l)2024
2270 y Fv(2)2063 2255 y Fw(j)28 b Fy(>)2266 2187 y(\015)p
2232 2232 124 4 v 2232 2323 a Fw(j)p Fy(l)2289 2338 y
Fv(2)2328 2323 y Fw(j)2383 2255 y Fy(;)44 b Fw(8)p Fy(l)2538
2270 y Fv(1)2578 2255 y Fy(;)17 b(l)2651 2270 y Fv(2)2718
2255 y Fw(2)28 b Ft(Z)22 b Fw(n)g(f)p FA(0)p Fw(g)3121
2144 y Fz(o)100 2486 y Fo(wher)-5 b(e)34 b FA(0)27 b
Fy(<)h(\015)33 b(<)27 b FA(1)p Fy(=)p FA(6)p Fo(.)199
2607 y(Ther)-5 b(e)30 b(exist)g(c)-5 b(onstants)p 1121
2552 60 4 v 30 w Fy(\033)32 b(>)27 b FA(0)p Fo(,)p 1421
2552 46 4 v 31 w Fy(s)h(>)g FA(2)p Fo(,)p 1709 2552 V
31 w Fy(")f(>)g FA(0)p Fo(,)p 1995 2527 77 4 v 31 w Fy(C)35
b(>)28 b FA(0)p Fo(,)i(such)h(that)g Fw(8)p Fy(")c Fw(2)h(B)3013
2622 y Fx(\015)3058 2607 y Fo(,)j Fw(j)p Fy(")p Fw(j)p
Fy(\015)3277 2571 y Fu(\000)p Fv(1)3398 2607 y Fw(\024)k
FA(\026)-55 b Fy(")o(=m)3683 2571 y Fv(2)3723 2607 y
Fo(,)100 2724 y(ther)-5 b(e)42 b(exists)h(a)f(classic)-5
b(al)41 b(solution)i Fy(u)p FA(\()p Fy(";)17 b(')p FA(\))41
b Fw(2)h(H)p 1999 2701 43 3 v 15 x Fx(\033)s(;)p 2062
2701 33 3 v(s)2141 2724 y Fo(of)g(\(1.4\))g(with)h FA(\()p
Fy(!)2834 2739 y Fv(1)2873 2724 y Fy(;)17 b(!)2978 2739
y Fv(2)3017 2724 y FA(\))42 b(=)g(\()p Fy(n=m;)17 b FA(1)27
b(+)h Fy(")p FA(\))100 2840 y Fo(satisfying)100 3069
y FA(\(1.5\))989 2955 y Fz(\014)989 3014 y(\014)989 3074
y(\014)1022 3069 y Fy(u)p FA(\()p Fy(";)17 b(')p FA(\))k
Fw(\000)h(j)p Fy(")p Fw(j)1618 2993 y Fq(1)p 1539 3005
189 3 v 1539 3046 a(2\()p Fn(d)p Fm(\000)p Fq(1\))1749
3069 y FA(\026)-56 b Fy(q)1785 3084 y Fx(")1822 3069
y FA(\()p Fy(')p FA(\))1962 2955 y Fz(\014)1962 3014
y(\014)1962 3074 y(\014)p 1995 3100 43 3 v 64 x Fx(\033)s(;)p
2058 3100 33 3 v(s)2122 3069 y Fw(\024)p 2227 2989 77
4 v 28 w Fy(C)2314 3002 y(m)2399 2966 y Fv(2)2439 3002
y Fw(j)p Fy(")p Fw(j)p 2314 3046 226 4 v 2339 3137 a
Fy(\015)21 b(!)2476 3103 y Fv(3)2472 3162 y(1)2550 3069
y Fw(j)p Fy(")p Fw(j)2740 2993 y Fq(1)p 2661 3005 189
3 v 2661 3046 a(2\()p Fn(d)p Fm(\000)p Fq(1\))100 3300
y Fo(for)38 b(an)f(appr)-5 b(opriate)38 b(function)45
b FA(\026)-57 b Fy(q)1352 3315 y Fx(")1423 3300 y Fw(2)34
b(H)1607 3315 y Fx(\033)n(;s)1727 3300 y Fw(n)25 b(f)p
FA(0)p Fw(g)37 b Fo(of)h(the)g(form)45 b FA(\026)-56
b Fy(q)2553 3315 y Fx(")2590 3300 y FA(\()p Fy(')p FA(\))33
b(=)46 b(\026)-57 b Fy(q)2920 3315 y Fv(+)2980 3300 y
FA(\()p Fy(')3082 3315 y Fv(2)3121 3300 y FA(\)+)45 b(\026)-56
b Fy(q)3316 3315 y Fu(\000)3375 3300 y FA(\(2)p Fy(m')3611
3315 y Fv(1)3675 3300 y Fw(\000)100 3416 y Fy(n')222
3431 y Fv(2)261 3416 y FA(\))p Fo(.)199 3533 y(As)51
b(a)f(c)-5 b(onse)g(quenc)g(e,)53 b(e)-5 b(quation)51
b(\(1.1\))e(admits)i(the)f(quasi-p)-5 b(erio)g(dic)49
b(solution)i Fy(v)t FA(\()p Fy(";)17 b(t;)g(x)p FA(\))56
b(:=)100 3649 y Fy(u)p FA(\()p Fy(";)17 b(!)345 3664
y Fv(1)383 3649 y Fy(t;)g(x)p FA(+)p Fy(!)654 3664 y
Fv(2)694 3649 y Fy(t)p FA(\))25 b Fo(with)g(two)g(fr)-5
b(e)g(quencies)24 b FA(\()p Fy(!)1752 3664 y Fv(1)1791
3649 y Fy(;)17 b(!)1896 3664 y Fv(2)1935 3649 y FA(\))28
b(=)d(\()p Fy(n=m;)17 b FA(1+)p Fy(")p FA(\))25 b Fo(and)f(the)h(map)g
Fy(t)j Fw(!)f Fy(v)t FA(\()p Fy(";)17 b(t;)g Fw(\001)p
FA(\))26 b Fw(2)100 3765 y Fy(H)p 189 3691 47 3 v 189
3729 a Fx(\033)235 3765 y FA(\()p Ft(T)p FA(\))35 b Fo(has)g(the)f
(form)949 3726 y Fv(4)109 3885 y Fz(\014)109 3945 y(\014)109
4005 y(\014)142 4000 y Fy(v)t FA(\()p Fy(";)17 b(t;)g(x)p
FA(\))k Fw(\000)i(j)p Fy(")p Fw(j)804 3923 y Fq(1)p 725
3935 189 3 v 725 3977 a(2\()p Fn(d)p Fm(\000)p Fq(1\))928
3889 y Fz(h)982 4000 y FA(\026)-56 b Fy(q)1018 4015 y
Fv(+)1077 4000 y FA(\()p Fy(x)23 b FA(+)f(\(1)g(+)g Fy(")p
FA(\))p Fy(t)p FA(\))g(+)29 b(\026)-56 b Fy(q)1818 4015
y Fu(\000)1877 4000 y FA(\(\(1)22 b Fw(\000)g Fy(")p
FA(\))p Fy(nt)h Fw(\000)f Fy(nx)p FA(\))2573 3889 y Fz(i)2621
3885 y(\014)2621 3945 y(\014)2621 4005 y(\014)2654 4069
y Fx(H)p 2717 4021 42 3 v 2717 4050 a Fn(\033)2759 4069
y Fv(\()p Fp(T)p Fv(\))2890 4000 y FA(=)27 b Fy(O)3071
3889 y Fz(\020)3166 3932 y Fy(m)3251 3896 y Fv(2)p 3140
3977 177 4 v 3140 4068 a Fy(\015)22 b(!)3278 4034 y Fv(3)3274
4093 y(1)3326 4000 y Fw(j)p Fy(")p Fw(j)3461 3923 y Fq(2)p
Fn(d)p Fm(\000)p Fq(1)p 3437 3936 189 3 v 3437 3977 a(2\()p
Fn(d)p Fm(\000)p Fq(1\))3640 3889 y Fz(\021)3716 4000
y Fy(:)100 4343 y FA(A)m(t)30 b(the)g(\014rst)g(order)g(the)h(quasi-p)s
(erio)s(dic)f(solution)h Fy(v)t FA(\()p Fy(";)17 b(t;)g(x)p
FA(\))29 b(of)g(equation)i(\(1.1\))e(is)h(the)h(sup)s(erp)s(o-)100
4459 y(sition)37 b(of)f(t)m(w)m(o)i(w)m(a)m(v)m(es)h(tra)m(v)m(eling)f
(in)f(opp)s(osite)h(directions)g(\(in)f(general,)h(b)s(oth)f(comp)s
(onen)m(ts)h Fy(q)3666 4474 y Fv(+)3725 4459 y FA(,)100
4575 y Fy(q)143 4590 y Fu(\000)234 4575 y FA(are)33 b(non)g(trivial\).)
199 4692 y(The)h(bifurcation)f(of)f(quasi-p)s(erio)s(dic)h(solutions)h
(lo)s(oks)f(quite)g(di\013eren)m(t)h(if)e Fy(!)3072 4707
y Fv(1)3144 4692 y FA(is)h(irrational.)100 4871 y FG(Theorem)25
b(B.)d Fo(L)-5 b(et)25 b Fy(!)914 4886 y Fv(1)981 4871
y Fw(2)j Ft(R)p Fw(n)p Ft(Q)p Fo(.)42 b(Assume)24 b(that)i
Fy(f)35 b Fo(satis\014es)25 b(assumption)f FA(\()p FG(H)p
FA(\))p Fo(,)3052 4791 y Fz(R)3118 4817 y Fv(2)p Fx(\031)3099
4906 y Fv(0)3217 4871 y Fy(a)3268 4886 y Fv(2)p Fx(d)p
Fu(\000)p Fv(1)3434 4871 y FA(\()p Fy(')3536 4886 y Fv(1)3576
4871 y FA(\))p Fy(d')3729 4886 y Fv(1)100 4989 y Fw(6)p
FA(=)j(0)35 b Fo(and)f Fy(f)11 b FA(\()p Fy(')637 5004
y Fv(1)676 4989 y Fy(;)17 b(u)p FA(\))27 b Fw(2)h Fy(H)1024
4953 y Fx(s)1060 4989 y FA(\()p Ft(T)p FA(\))p Fo(,)35
b Fy(s)28 b Fw(\025)g FA(1)p Fo(,)35 b(for)f(al)5 b(l)35
b Fy(u)p Fo(.)p 100 5105 499 4 v 199 5171 a Fv(2)235
5203 y FF(Giv)n(en)27 b Fl(z)f Fk(2)e Fj(C)p FF(,)j Fl(z)772
5173 y Fi(\003)837 5203 y FF(denotes)h(its)g(complex)f(conjugate.)199
5276 y Fv(3)235 5308 y FF(The)g(pro)r(of)g(that)h Fk(B)857
5320 y Fh(\015)918 5308 y Fk(\\)18 b FF(\(0)p Fl(;)c(")1141
5320 y FB(0)1178 5308 y FF(\))28 b(and)g Fk(B)1455 5320
y Fh(\015)1515 5308 y Fk(\\)19 b FF(\()p Fk(\000)p Fl(")1725
5320 y FB(0)1762 5308 y Fl(;)14 b FF(0\))27 b(are)g(b)r(oth)h(uncoun)n
(table)f Fk(8)p Fl(")2788 5320 y FB(0)2848 5308 y Fl(>)c
FF(0)k(is)g(lik)n(e)h(in)f([BP].)199 5381 y Fv(4)235
5413 y FF(W)-7 b(e)28 b(denote)f Fl(H)721 5383 y Fh(\033)766
5413 y FF(\()p Fj(T)p FF(\))c(:=)g Fk(f)p Fl(u)p FF(\()p
Fl(')p FF(\))g(=)1338 5351 y Fg(P)1425 5438 y Fh(l)p
Fi(2)p Ff(Z)1554 5413 y FF(^)-48 b Fl(u)1596 5425 y Fh(l)1621
5413 y Fl(e)1660 5383 y FB(i)p Fh(l')1799 5413 y FF(:)56
b(^)-48 b Fl(u)1920 5383 y Fi(\003)1920 5437 y Fh(l)1981
5413 y FF(=)28 b(^)-47 b Fl(u)2117 5425 y Fi(\000)p Fh(l)2208
5413 y Fl(;)41 b Fk(j)p Fl(u)p Fk(j)2366 5428 y Fh(H)2424
5411 y Fe(\033)2464 5428 y FB(\()p Ff(T)p FB(\))2582
5413 y FF(:=)2693 5351 y Fg(P)2780 5438 y Fh(l)p Fi(2)p
Ff(Z)2903 5413 y Fk(j)5 b FF(^)-47 b Fl(u)2974 5425 y
Fh(l)2999 5413 y Fk(j)p Fl(e)3061 5383 y Fh(\033)r Fi(j)p
Fh(l)p Fi(j)3189 5413 y Fl(<)23 b FF(+)p Fk(1g)p FF(.)p
eop end
%%Page: 4 4
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b(et)35 b Fw(C)419 281 y Fx(\015)492 266 y Fw(\032)28
b Fy(D)j Fw(\021)d FA(\()p Fw(\000)p Fy(")975 281 y Fv(0)1014
266 y Fy(;)17 b(")1104 281 y Fv(0)1143 266 y FA(\))22
b Fw(\002)h FA(\(1)p Fy(;)17 b FA(2\))34 b Fo(b)-5 b(e)34
b(the)h(unc)-5 b(ountable)35 b(zer)-5 b(o-me)g(asur)g(e)33
b(Cantor)i(set)3404 227 y Fv(5)100 600 y FA(\(1.6\))133
b Fw(C)486 615 y Fx(\015)558 600 y FA(:=)689 366 y Fz(8)689
456 y(>)689 485 y(<)689 665 y(>)689 695 y(:)794 463 y
FA(\()p Fy(";)17 b(!)983 478 y Fv(1)1022 463 y FA(\))27
b Fw(2)h Fy(D)j FA(:)55 b Fy(!)1436 478 y Fv(1)1515 463
y Fy(=)-61 b Fw(2)28 b Ft(Q)17 b Fy(;)1845 396 y(!)1906
411 y Fv(1)p 1845 440 101 4 v 1845 531 a Fy(!)1906 546
y Fv(2)1994 463 y Fy(=)-61 b Fw(2)29 b Ft(Q)17 b Fy(;)116
b Fw(j)p Fy(!)2403 478 y Fv(1)2442 463 y Fy(l)2471 478
y Fv(1)2532 463 y FA(+)22 b Fy("l)2705 478 y Fv(2)2745
463 y Fw(j)27 b Fy(>)3069 396 y(\015)p 2913 440 368 4
v 2913 531 a Fw(j)p Fy(l)2970 546 y Fv(1)3010 531 y Fw(j)21
b FA(+)h Fw(j)p Fy(l)3214 546 y Fv(2)3254 531 y Fw(j)3308
463 y Fy(;)1375 704 y Fw(j)p Fy(!)1464 719 y Fv(1)1503
704 y Fy(l)1532 719 y Fv(1)1594 704 y FA(+)g(\(2)g(+)g
Fy(")p FA(\))p Fy(l)2012 719 y Fv(2)2051 704 y Fw(j)27
b Fy(>)2375 637 y(\015)p 2220 681 V 2220 772 a Fw(j)p
Fy(l)2277 787 y Fv(1)2316 772 y Fw(j)22 b FA(+)g Fw(j)p
Fy(l)2521 787 y Fv(2)2560 772 y Fw(j)2614 704 y Fy(;)52
b Fw(8)28 b Fy(l)2805 719 y Fv(1)2844 704 y Fy(;)17 b(l)2917
719 y Fv(2)2984 704 y Fw(2)28 b Ft(Z)23 b Fw(n)f(f)p
FA(0)p Fw(g)3487 366 y Fz(9)3487 456 y(>)3487 485 y(=)3487
665 y(>)3487 695 y(;)3592 600 y Fy(:)100 945 y Fo(Fix)45
b(any)h FA(0)h Fy(<)p 702 890 46 4 v 48 w(s)h(<)g(s)30
b Fw(\000)h FA(1)p Fy(=)p FA(2)p Fo(.)77 b(Ther)-5 b(e)45
b(exist)h(p)-5 b(ositive)45 b(c)-5 b(onstants)p 2688
890 V 45 w Fy(")p Fo(,)p 2812 865 77 4 v 48 w Fy(C)7
b Fo(,)p 2968 890 60 4 v 49 w Fy(\033)52 b(>)47 b FA(0)p
Fo(,)i(such)c(that,)100 1075 y Fw(8)p FA(\()p Fy(";)17
b(!)344 1090 y Fv(1)382 1075 y FA(\))43 b Fw(2)f(C)623
1090 y Fx(\015)711 1075 y Fo(with)h Fw(j)p Fy(")p Fw(j)p
Fy(\015)1089 1038 y Fu(\000)p Fv(1)1224 1075 y Fy(<)p
1342 1020 46 4 v 42 w(")g Fo(and)58 b Fy(")1707 994 y
Fz(R)1773 1021 y Fv(2)p Fx(\031)1754 1109 y Fv(0)1872
1075 y Fy(a)1923 1090 y Fv(2)p Fx(d)p Fu(\000)p Fv(1)2089
1075 y FA(\()p Fy(')2191 1090 y Fv(1)2230 1075 y FA(\))17
b Fy(d')2400 1090 y Fv(1)2481 1075 y Fy(>)42 b FA(0)p
Fo(,)j(ther)-5 b(e)42 b(exists)h(a)f(nontrivial)100 1192
y(solution)34 b Fy(u)p FA(\()p Fy(";)17 b(')p FA(\))27
b Fw(2)h(H)p 959 1170 43 3 v 15 x Fx(\033)s(;)p 1022
1170 33 3 v(s)1093 1192 y Fo(of)34 b(e)-5 b(quation)35
b(\(1.4\))f(with)h FA(\()p Fy(!)2151 1207 y Fv(1)2190
1192 y Fy(;)17 b(!)2295 1207 y Fv(2)2334 1192 y FA(\))27
b(=)h(\()p Fy(!)2602 1207 y Fv(1)2641 1192 y Fy(;)17
b FA(1)22 b(+)g Fy(")p FA(\))34 b Fo(satisfying)100 1421
y FA(\(1.7\))1014 1307 y Fz(\014)1014 1367 y(\014)1014
1426 y(\014)1047 1421 y Fy(u)p FA(\()p Fy(";)17 b(')p
FA(\))k Fw(\000)h(j)p Fy(")p Fw(j)1644 1345 y Fq(1)p
1564 1357 189 3 v 1564 1398 a(2\()p Fn(d)p Fm(\000)p
Fq(1\))1775 1421 y FA(\026)-57 b Fy(q)1810 1436 y Fx(")1847
1421 y FA(\()p Fy(')1949 1436 y Fv(2)1989 1421 y FA(\))2027
1307 y Fz(\014)2027 1367 y(\014)2027 1426 y(\014)p 2060
1453 43 3 v 64 x Fx(\033)s(;)p 2123 1453 33 3 v(s)2187
1421 y Fw(\024)p 2327 1341 77 4 v 63 w Fy(C)2414 1354
y Fw(j)p Fy(")p Fw(j)p 2414 1399 101 4 v 2436 1490 a
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2635 1357 189 3 v 2635 1398 a(2\()p Fn(d)p Fm(\000)p
Fq(1\))100 1648 y Fo(for)34 b(some)g(function)42 b FA(\026)-56
b Fy(q)931 1663 y Fx(")968 1648 y FA(\()p Fy(')1070 1663
y Fv(2)1109 1648 y FA(\))28 b Fw(2)g Fy(H)p 1358 1574
47 3 v 1358 1612 a Fx(\033)1404 1648 y FA(\()p Ft(T)p
FA(\))23 b Fw(n)f(f)p FA(0)p Fw(g)p Fo(.)199 1764 y(As)j(a)g(c)-5
b(onse)g(quenc)g(e,)26 b(e)-5 b(quation)24 b(\(1.1\))g(admits)h(the)g
(non-trivial)f(quasi-p)-5 b(erio)g(dic)23 b(solution)i
Fy(v)t FA(\()p Fy(";)17 b(t;)g(x)p FA(\))27 b(:=)100
1881 y Fy(u)p FA(\()p Fy(";)17 b(!)345 1896 y Fv(1)383
1881 y Fy(t;)g(x)12 b FA(+)g Fy(!)678 1896 y Fv(2)717
1881 y Fy(t)p FA(\))30 b Fo(with)g(two)g(fr)-5 b(e)g(quencies)30
b FA(\()p Fy(!)1796 1896 y Fv(1)1835 1881 y Fy(;)17 b(!)1940
1896 y Fv(2)1979 1881 y FA(\))27 b(=)j(\()p Fy(!)2249
1896 y Fv(1)2288 1881 y Fy(;)17 b FA(1)12 b(+)g Fy(")p
FA(\))29 b Fo(and)h(the)g(map)g Fy(t)e Fw(!)f Fy(v)t
FA(\()p Fy(";)17 b(t;)g Fw(\001)p FA(\))26 b Fw(2)100
1997 y Fy(H)p 189 1923 V 189 1961 a Fx(\033)235 1997
y FA(\()p Ft(T)p FA(\))35 b Fo(has)g(the)f(form)658 2081
y Fz(\014)658 2141 y(\014)658 2201 y(\014)691 2196 y
Fy(v)t FA(\()p Fy(";)17 b(t;)g(x)p FA(\))22 b Fw(\000)g(j)p
Fy(")p Fw(j)1353 2119 y Fq(1)p 1274 2131 189 3 v 1274
2172 a(2\()p Fn(d)p Fm(\000)p Fq(1\))1484 2196 y FA(\026)-56
b Fy(q)1520 2211 y Fx(")1557 2196 y FA(\()p Fy(x)23 b
FA(+)f(\(1)f(+)h Fy(")p FA(\))p Fy(t)p FA(\))2134 2081
y Fz(\014)2134 2141 y(\014)2134 2201 y(\014)2167 2265
y Fx(H)p 2230 2217 42 3 v 2230 2246 a Fn(\033)2272 2265
y Fv(\()p Fp(T)p Fv(\))2403 2196 y FA(=)28 b Fy(O)2585
2085 y Fz(\020)2643 2196 y Fy(\015)2699 2155 y Fu(\000)p
Fv(1)2794 2196 y Fw(j)p Fy(")p Fw(j)2929 2118 y Fq(2)p
Fn(d)p Fm(\000)p Fq(1)p 2905 2131 189 3 v 2905 2172 a(2\()p
Fn(d)p Fm(\000)p Fq(1\))3108 2085 y Fz(\021)3167 2196
y Fy(:)100 2543 y FG(Remark)35 b(1.)40 b Fo(Imp)-5 b(osing)31
b(in)i(the)f(de\014nition)g(of)h Fw(C)1939 2558 y Fx(\015)2017
2543 y Fo(the)f(c)-5 b(ondition)32 b Fy(!)2661 2558 y
Fv(1)2700 2543 y Fy(=!)2810 2558 y Fv(2)2877 2543 y FA(=)c
Fy(!)3042 2558 y Fv(1)3081 2543 y Fy(=)p FA(\(1)17 b(+)g
Fy(")p FA(\))28 b Fw(2)g Ft(Q)33 b Fo(we)100 2659 y(obtain,)39
b(by)h(The)-5 b(or)g(em)38 b(B)h(the)g(existenc)-5 b(e)38
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3333 a(2)1406 3265 y(\()p Fy(@)1495 3280 y Fx(')1541
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Fx(')2237 3289 y Fq(2)2275 3265 y Fy(u)p FA(\))631 3535
y(\000\()p Fy(u;)17 b(\016)t FA(\))82 b(:=)1183 3399
y Fz(Z)1238 3625 y Fp(T)1282 3606 y Fq(2)1348 3467 y
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3603 a(2)1648 3535 y(\()p Fy(@)1737 3550 y Fx(')1783
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3535 y FA(+)g(\()p Fy(@)2164 3550 y Fx(')2210 3559 y
Fq(1)2249 3535 y Fy(u)p FA(\)\()p Fy(@)2432 3550 y Fx(')2478
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3948 y(inspired)34 b(to)e([BB1]-[BB2],)g(see)i(also)f([AB].)100
4143 y(2.1.)48 b FG(The)25 b(V)-9 b(ariational)26 b(Ly)m(apuno)m(v-Sc)m
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4259 y(\011)176 4274 y Fv(0)243 4259 y FA(:)k Fw(H)381
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y Fz(X)1086 5048 y Fx(l)q Fu(2)p Fp(Z)1199 5029 y Fq(2)1257
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4884 y Fv(i)p Fx(l)q Fu(\001)p Fx(')1504 4925 y Fw(2)28
b(H)1682 4940 y Fx(\033)n(;s)1809 4925 y Fw(j)72 b FA(^)-56
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y Fv(1)2660 4925 y FA(+)22 b(2)p Fy(l)2836 4940 y Fv(2)2875
4925 y FA(\))28 b Fw(6)p FA(=)g(0)3094 4815 y Fz(o)3192
4925 y Fy(:)100 5180 y FA(In)i(view)i(of)d(the)i(v)-5
b(ariational)29 b(argumen)m(t)i(that)f(w)m(e)h(shall)g(use)g(to)f(solv)
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Fy(Q)e FA(as)1505 5413 y Fy(Q)c FA(=)g Fy(Q)1791 5428
y Fv(+)1872 5413 y Fw(\010)23 b Fy(Q)2049 5428 y Fv(0)2111
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%%Page: 7 7
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b(7)100 266 y FA(where)348 227 y Fv(6)200 431 y Fy(Q)277
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Fx(l)1162 431 y FA(=)28 b(0)k(for)g Fy(l)42 b(=)-61 b
Fw(2)28 b FA(\003)1717 446 y Fv(+)1776 321 y Fz(o)1870
431 y FA(=)1973 321 y Fz(n)2040 431 y Fy(q)2083 446 y
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Ft(T)p FA(\))2981 321 y Fz(o)220 648 y Fy(Q)297 663 y
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y Fy(Q)277 880 y Fu(\000)419 865 y FA(:=)605 754 y Fz(n)672
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Fy(')2779 880 y Fv(2)2818 865 y FA(\))p Fy(;)49 b(q)2975
880 y Fu(\000)3034 865 y FA(\()p Fw(\001)p FA(\))28 b
Fw(2)g Fy(H)3349 817 y Fx(\033)n(;s)3341 889 y Fv(0)3443
865 y FA(\()p Ft(T)p FA(\))3585 754 y Fz(o)100 1050 y
FA(and)100 1215 y(\(2.2\))154 b(\003)523 1230 y Fv(+)609
1215 y FA(:=)740 1105 y Fz(n)806 1215 y Fy(l)30 b Fw(2)e
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1230 y Fv(1)1309 1215 y FA(=)28 b(0)p Fy(;)49 b(l)29
b Fw(6)p FA(=)f(0)1749 1105 y Fz(o)1815 1215 y Fy(;)114
b FA(\003)2024 1230 y Fu(\000)2111 1215 y FA(:=)2241
1105 y Fz(n)2308 1215 y Fy(l)30 b Fw(2)e Ft(Z)2527 1174
y Fv(2)2627 1215 y FA(:)60 b Fy(l)2743 1230 y Fv(1)2805
1215 y FA(+)22 b(2)p Fy(l)2981 1230 y Fv(2)3048 1215
y FA(=)27 b(0)p Fy(;)49 b(l)30 b Fw(6)p FA(=)e(0)3488
1105 y Fz(o)3571 1215 y Fy(:)100 1400 y FA(W)-8 b(e)33
b(shall)g(also)f(use)i(in)f Fy(Q)g FA(the)g(norm)660
1575 y Fw(j)p Fy(q)t Fw(j)763 1534 y Fv(2)763 1601 y
Fx(H)826 1582 y Fq(1)892 1575 y FA(=)27 b Fw(j)p Fy(q)1066
1590 y Fv(+)1125 1575 y Fw(j)1153 1534 y Fv(2)1153 1601
y Fx(H)1216 1582 y Fq(1)1251 1601 y Fv(\()p Fp(T)p Fv(\))1376
1575 y FA(+)22 b Fy(q)1521 1534 y Fv(2)1517 1600 y(0)1582
1575 y FA(+)g Fw(j)p Fy(q)1751 1590 y Fu(\000)1810 1575
y Fw(j)1838 1534 y Fv(2)1838 1601 y Fx(H)1901 1582 y
Fq(1)1936 1601 y Fv(\()p Fp(T)p Fv(\))2083 1575 y Fw(\030)2368
1481 y Fz(X)2205 1697 y Fx(l)q Fu(2)p Fv(\003)2323 1706
y Fm(\000)2375 1697 y Fu([f)p Fv(0)p Fu(g[)p Fv(\003)2623
1706 y Fq(+)2699 1575 y FA(^)-57 b Fy(q)2738 1534 y Fv(2)2734
1600 y Fx(l)2778 1575 y FA(\()p Fw(j)p Fy(l)r Fw(j)2903
1534 y Fv(2)2964 1575 y FA(+)22 b(1\))17 b Fy(:)100 1843
y FA(W)-8 b(e)33 b(decomp)s(ose)h(the)f(space)g Fw(H)1271
1858 y Fx(\033)n(;s)1394 1843 y FA(=)28 b Fy(Q)22 b Fw(\010)h
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y Fz(n)969 2037 y Fy(p)28 b FA(=)1152 1942 y Fz(X)1150
2159 y Fx(l)q Fu(2)p Fp(Z)1263 2140 y Fq(2)1322 2037
y FA(^)-57 b Fy(p)1363 2052 y Fx(l)1389 2037 y Fy(e)1434
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b(H)1751 2052 y Fx(\033)n(;s)1879 2037 y Fw(j)73 b FA(^)-57
b Fy(p)2021 2052 y Fx(l)2074 2037 y FA(=)28 b(0)97 b(for)g
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2052 y Fv(2)2784 2037 y FA(+)22 b Fy(l)2911 2052 y Fv(1)2951
2037 y FA(\))27 b(=)h(0)3169 1926 y Fz(o)3267 2037 y
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FA(,)30 b(setting)h Fy(u)d FA(=)f Fy(q)21 b FA(+)d Fy(p)27
b Fw(2)h(H)3657 2314 y Fx(\033)n(;s)100 2416 y FA(with)33
b Fy(q)e Fw(2)d Fy(Q)33 b FA(and)g Fy(p)28 b Fw(2)g Fy(P)14
b FA(,)32 b(w)m(e)h(obtain)1016 2503 y Fz(\032)1107 2561
y Fy(L)1173 2576 y Fv(1)1213 2561 y FA([)p Fy(q)t FA(])22
b(+)55 b(\005)1540 2576 y Fx(Q)1599 2561 y Fy(f)11 b
FA(\()p Fy(')1760 2576 y Fv(1)1799 2561 y Fy(;)17 b(q)26
b FA(+)c Fy(p;)17 b(\016)t FA(\))27 b(=)h(0)292 b(\()p
Fy(Q)p FA(\))1107 2712 y Fw(L)1176 2727 y Fx(")1213 2712
y FA([)p Fy(p)p FA(])22 b(+)g Fy(")p FA(\005)1555 2727
y Fx(P)1613 2712 y Fy(f)11 b FA(\()p Fy(')1774 2727 y
Fv(1)1813 2712 y Fy(;)17 b(q)26 b FA(+)c Fy(p;)17 b(\016)t
FA(\))27 b(=)h(0)292 b(\()p Fy(P)14 b FA(\))100 2871
y(where)33 b(\005)454 2886 y Fx(Q)542 2871 y FA(:)28
b Fw(H)681 2886 y Fx(\033)n(;s)804 2871 y Fw(!)f Fy(Q)p
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Fw(H)1366 2886 y Fx(\033)n(;s)1489 2871 y Fw(!)f Fy(P)46
b FA(are)33 b(the)g(pro)5 b(jectors)33 b(resp)s(ectiv)m(ely)j(on)m(to)d
Fy(Q)f FA(and)h Fy(P)14 b FA(.)199 3011 y(In)30 b(order)f(to)f(pro)m(v)
m(e)j(analiticit)m(y)f(of)e(the)i(solutions)g(and)f(to)f(highligh)m(t)i
(the)f(compactness)j(of)c(the)100 3127 y(problem)f(w)m(e)g(p)s(erform)f
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3362 y Fy(Q)h FA(=)g Fy(Q)1930 3377 y Fv(1)1992 3362
y Fw(\010)22 b Fy(Q)2168 3377 y Fv(2)100 3500 y FA(where)105
3666 y Fy(Q)182 3681 y Fv(1)249 3666 y FA(:=)28 b Fy(Q)457
3681 y Fv(1)497 3666 y FA(\()p Fy(N)10 b FA(\))28 b(:=)819
3555 y Fz(n)918 3666 y Fy(q)k FA(=)1114 3571 y Fz(X)1096
3787 y Fu(j)p Fx(l)q Fu(j\024)p Fx(N)1299 3666 y FA(^)-56
b Fy(q)1335 3681 y Fx(l)1361 3666 y Fy(e)1406 3625 y
Fv(i)p Fx(l)q Fu(\001)p Fx(')1545 3666 y Fw(2)28 b Fy(Q)1716
3555 y Fz(o)1800 3666 y Fy(;)49 b(Q)1953 3681 y Fv(2)2020
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Fy(q)j FA(=)2852 3571 y Fz(X)2835 3787 y Fu(j)p Fx(l)q
Fu(j)p Fx(>N)3038 3666 y FA(^)-57 b Fy(q)3073 3681 y
Fx(l)3100 3666 y Fy(e)3145 3625 y Fv(i)p Fx(l)q Fu(\001)p
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3666 y Fy(:)100 3934 y FA(Setting)33 b Fy(q)e FA(=)d
Fy(q)657 3949 y Fv(1)719 3934 y FA(+)22 b Fy(q)860 3949
y Fv(2)932 3934 y FA(with)33 b Fy(q)1197 3949 y Fv(1)1264
3934 y Fw(2)28 b Fy(Q)1435 3949 y Fv(1)1508 3934 y FA(and)k
Fy(q)1740 3949 y Fv(2)1808 3934 y Fw(2)c Fy(Q)1979 3949
y Fv(2)2018 3934 y FA(,)33 b(w)m(e)h(\014nally)f(get)361
4119 y Fy(L)427 4134 y Fv(1)467 4119 y FA([)p Fy(q)537
4134 y Fv(1)577 4119 y FA(])22 b(+)g(\005)797 4134 y
Fx(Q)853 4143 y Fq(1)891 4008 y Fz(h)938 4119 y Fy(f)11
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4119 y FA(+)g Fy(p;)17 b(\016)t FA(\))1766 4008 y Fz(i)1840
4119 y FA(=)27 b(0)111 b Fw(\()-17 b(\))110 b Fy(d)p
FA(\011)2523 4134 y Fx(")2560 4119 y FA(\()p Fy(u)p FA(\)[)p
Fy(h)p FA(])27 b(=)g(0)65 b Fw(8)p Fy(h)28 b Fw(2)h Fy(Q)3357
4134 y Fv(1)3494 4119 y FA(\()p Fy(Q)3609 4134 y Fv(1)3648
4119 y FA(\))-3586 b(\(2.3\))361 4336 y Fy(L)427 4351
y Fv(1)467 4336 y FA([)p Fy(q)537 4351 y Fv(2)577 4336
y FA(])22 b(+)g(\005)797 4351 y Fx(Q)853 4360 y Fq(2)891
4225 y Fz(h)938 4336 y Fy(f)11 b FA(\()p Fy(')1099 4351
y Fv(1)1138 4336 y Fy(;)17 b(q)1225 4351 y Fv(1)1287
4336 y FA(+)22 b Fy(q)1428 4351 y Fv(2)1490 4336 y FA(+)g
Fy(p;)17 b(\016)t FA(\))1766 4225 y Fz(i)1840 4336 y
FA(=)27 b(0)111 b Fw(\()-17 b(\))110 b Fy(d)p FA(\011)2523
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FA(])27 b(=)g(0)65 b Fw(8)p Fy(h)28 b Fw(2)h Fy(Q)3357
4351 y Fv(2)3494 4336 y FA(\()p Fy(Q)3609 4351 y Fv(2)3648
4336 y FA(\))-3586 b(\(2.4\))385 4552 y Fw(L)454 4567
y Fx(")490 4552 y FA([)p Fy(p)p FA(])23 b(+)f Fy(")p
FA(\005)833 4567 y Fx(P)891 4442 y Fz(h)938 4552 y Fy(f)11
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4552 y FA(=)27 b(0)111 b Fw(\()-17 b(\))110 b Fy(d)p
FA(\011)2523 4567 y Fx(")2560 4552 y FA(\()p Fy(u)p FA(\)[)p
Fy(h)p FA(])27 b(=)g(0)65 b Fw(8)p Fy(h)28 b Fw(2)h Fy(P)143
b FA(\()p Fy(P)14 b FA(\))-3539 b(\(2.5\))100 4743 y(where)33
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4758 y Fx(i)1095 4743 y FA(are)33 b(the)g(pro)5 b(jectors)33
b(on)m(to)g Fy(Q)2181 4758 y Fx(i)2242 4743 y FA(\()p
Fy(i)28 b FA(=)f(1)p Fy(;)17 b FA(2\).)199 4883 y(W)-8
b(e)38 b(shall)f(solv)m(e)i(\014rst)e(the)h(\()p Fy(Q)1341
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%%Page: 8 8
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end
%%Page: 9 9
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1540 y Fw(\024)2651 1473 y Fy(C)2721 1488 y Fv(0)2760
1473 y FA(\()p Fy(R)q FA(\))p 2651 1518 V 2717 1609 a
Fy(N)2805 1580 y Fv(2)2921 1540 y Fw(j)p Fy(h)p Fw(j)3033
1557 y Fx(H)3096 1538 y Fq(1)3234 1540 y Fw(8)p Fy(h)g
Fw(2)g Fy(Q)3544 1555 y Fv(1)3600 1540 y Fy(:)100 1769
y Fo(Pr)-5 b(o)g(of.)41 b FA(In)33 b(the)g(App)s(endix.)2547
b Fd(\003)100 2118 y FA(2.3.)48 b FG(The)43 b FA(\()p
Fy(Q)649 2133 y Fv(1)689 2118 y FA(\))p FG(-equation.)50
b FA(Once)39 b(the)e(\()p Fy(Q)1806 2133 y Fv(2)1846
2118 y FA(\)-\()p Fy(P)14 b FA(\)-equations)37 b(ha)m(v)m(e)i(b)s(een)f
(solv)m(ed)h(b)m(y)f(\()p Fy(q)3527 2133 y Fv(2)3567
2118 y FA(\()p Fy(q)3648 2133 y Fv(1)3688 2118 y FA(\))p
Fy(;)100 2234 y(p)p FA(\()p Fy(q)230 2249 y Fv(1)269
2234 y FA(\)\))28 b Fw(2)33 b Fy(Q)549 2249 y Fv(2)611
2234 y Fw(\010)22 b Fy(P)46 b FA(there)33 b(remains)h(the)f(\014nite)g
(dimensional)h(\()p Fy(Q)2501 2249 y Fv(1)2541 2234 y
FA(\)-equation)100 2411 y(\(2.12\))619 b Fy(L)1035 2426
y Fv(1)1075 2411 y FA([)p Fy(q)1145 2426 y Fv(1)1185
2411 y FA(])22 b(+)g(\005)1405 2426 y Fx(Q)1461 2435
y Fq(1)1499 2411 y Fy(f)11 b FA(\()p Fy(')1660 2426 y
Fv(1)1699 2411 y Fy(;)17 b(q)1786 2426 y Fv(1)1848 2411
y FA(+)22 b Fy(q)1989 2426 y Fv(2)2029 2411 y FA(\()p
Fy(q)2110 2426 y Fv(1)2149 2411 y FA(\))g(+)g Fy(p)p
FA(\()p Fy(q)2437 2426 y Fv(1)2477 2411 y FA(\))p Fy(;)17
b(\016)t FA(\))27 b(=)h(0)k Fy(:)199 2648 y FA(The)45
b(geometric)g(in)m(terpretation)g(of)f(the)h(construction)g(of)f(\()p
Fy(q)2576 2663 y Fv(2)2615 2648 y FA(\()p Fy(q)2696 2663
y Fv(1)2736 2648 y FA(\),)j Fy(p)p FA(\()p Fy(q)2978
2663 y Fv(1)3017 2648 y FA(\)\))d(is)h(that)f(on)g(the)100
2765 y(\014nite)34 b(dimensional)h(sub-manifold)g Fy(Z)i
Fw(\021)d(f)p Fy(q)1781 2780 y Fv(1)1843 2765 y FA(+)23
b Fy(q)1985 2780 y Fv(2)2025 2765 y FA(\()p Fy(q)2106
2780 y Fv(1)2146 2765 y FA(\))f(+)h Fy(p)p FA(\()p Fy(q)2435
2780 y Fv(1)2475 2765 y FA(\))63 b(:)h Fw(j)p Fy(q)2738
2780 y Fv(1)2777 2765 y Fw(j)30 b Fy(<)g FA(2)p Fy(R)q
Fw(g)p FA(,)j(di\013eomorphic)100 2881 y(to)f(the)h(ball)1218
3017 y Fy(B)1292 3032 y Fv(2)p Fx(R)1413 3017 y FA(:=)28
b Fw(f)p Fy(q)1637 3032 y Fv(1)1704 3017 y Fw(2)g Fy(Q)1875
3032 y Fv(1)1975 3017 y FA(:)60 b Fw(j)p Fy(q)2133 3032
y Fv(1)2172 3017 y Fw(j)2200 3034 y Fx(H)2263 3015 y
Fq(1)2330 3017 y Fy(<)27 b FA(2)p Fy(R)q Fw(g)p Fy(;)100
3173 y FA(the)34 b(partial)f(deriv)-5 b(ativ)m(es)35
b(of)e(the)h(action)g(functional)f(\011)2179 3188 y Fx(")2249
3173 y FA(with)h(resp)s(ect)h(to)e(the)h(v)-5 b(ariables)34
b(\()p Fy(q)3582 3188 y Fv(2)3622 3173 y Fy(;)17 b(p)p
FA(\))100 3290 y(v)-5 b(anish.)41 b(W)-8 b(e)23 b(claim)h(that)f(at)g
(a)g(critical)h(p)s(oin)m(t)f(of)g(\011)1968 3305 y Fx(")2028
3290 y FA(restricted)h(to)f Fy(Z)7 b FA(,)25 b(also)e(the)h(partial)f
(deriv)-5 b(ativ)m(e)100 3406 y(of)36 b(\011)291 3421
y Fx(")364 3406 y FA(w.r.t.)57 b(the)38 b(v)-5 b(ariable)37
b Fy(q)1234 3421 y Fv(1)1310 3406 y FA(v)-5 b(anishes)39
b(and)e(therefore)g(that)g(suc)m(h)h(p)s(oin)m(t)f(is)h(critical)f
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3537 y Fx(")1490 3522 y FA(:)27 b Fw(H)1628 3537 y Fx(\033)n(;s)1751
3522 y Fw(!)h Ft(R)p FA(.)199 3638 y(Actually)40 b(the)g(bifurcation)f
(equation)h(\(2.12\))e(is)i(the)f(Euler-Lagrange)g(equation)h(of)e(the)
i(re-)100 3754 y(duced)33 b(Lagrangian)f(action)h(functional)537
3931 y(\010)607 3946 y Fx(";N)754 3931 y FA(:)28 b Fy(B)883
3946 y Fv(2)p Fx(R)1004 3931 y Fw(\032)g Fy(Q)1186 3946
y Fv(1)1253 3931 y Fw(!)g Ft(R)p Fy(;)309 b FA(\010)1859
3946 y Fx(";N)1979 3931 y FA(\()p Fy(q)2060 3946 y Fv(1)2099
3931 y FA(\))28 b(:=)g(\011)2372 3946 y Fx(")2408 3931
y FA(\()p Fy(q)2489 3946 y Fv(1)2551 3931 y FA(+)22 b
Fy(q)2692 3946 y Fv(2)2732 3931 y FA(\()p Fy(q)2813 3946
y Fv(1)2852 3931 y FA(\))g(+)g Fy(p)p FA(\()p Fy(q)3140
3946 y Fv(1)3180 3931 y FA(\)\))32 b Fy(:)100 4117 y
FG(Lemma)39 b(2.4.)i FA(\010)782 4132 y Fx(";N)930 4117
y Fw(2)35 b(C)1089 4081 y Fv(1)1129 4117 y FA(\()p Fy(B)1241
4132 y Fv(2)p Fx(R)1334 4117 y Fy(;)17 b Ft(R)p FA(\))34
b Fo(and)g(a)h(critic)-5 b(al)34 b(p)-5 b(oint)35 b Fy(q)2413
4132 y Fv(1)2480 4117 y Fw(2)28 b Fy(B)2648 4132 y Fv(2)p
Fx(R)2776 4117 y Fo(of)35 b FA(\010)2961 4132 y Fx(";N)3115
4117 y Fo(is)g(a)f(solution)h(of)100 4233 y(the)g(bifur)-5
b(c)g(ation)34 b(e)-5 b(quation)35 b(\(2.12\).)43 b(Mor)-5
b(e)g(over)35 b FA(\010)1967 4248 y Fx(";N)2122 4233
y Fo(c)-5 b(an)34 b(b)-5 b(e)34 b(written)h(as)100 4444
y FA(\(2.13\))731 b(\010)1151 4459 y Fx(";N)1271 4444
y FA(\()p Fy(q)1352 4459 y Fv(1)1392 4444 y FA(\))28
b(=)f(const)c(+)f Fy(")1950 4334 y Fz(\020)2009 4444
y FA(\000\()p Fy(q)2151 4459 y Fv(1)2191 4444 y FA(\))g(+)g
Fw(R)2433 4459 y Fx(";N)2553 4444 y FA(\()p Fy(q)2634
4459 y Fv(1)2673 4444 y FA(\))2711 4334 y Fz(\021)100
4647 y Fo(wher)-5 b(e)652 4825 y FA(\000\()p Fy(q)794
4840 y Fv(1)834 4825 y FA(\))27 b(:=)1030 4689 y Fz(Z)1085
4915 y Fp(T)1129 4896 y Fq(2)1195 4758 y FA(\(2)22 b(+)g
Fy(")p FA(\))p 1195 4802 291 4 v 1316 4893 a(2)1495 4825
y(\()p Fy(@)1584 4840 y Fx(')1630 4849 y Fq(2)1670 4825
y Fy(q)1713 4840 y Fv(1)1752 4825 y FA(\))1790 4784 y
Fv(2)1852 4825 y FA(+)g(\()p Fy(@)2039 4840 y Fx(')2085
4849 y Fq(1)2124 4825 y Fy(q)2167 4840 y Fv(1)2207 4825
y FA(\)\()p Fy(@)2334 4840 y Fx(')2380 4849 y Fq(2)2419
4825 y Fy(q)2462 4840 y Fv(1)2501 4825 y FA(\))h Fw(\000)f
Fy(a)2712 4840 y Fv(2)p Fx(d)p Fu(\000)p Fv(1)2878 4825
y FA(\()p Fy(')2980 4840 y Fv(1)3020 4825 y FA(\))3068
4758 y Fy(q)3115 4722 y Fv(2)p Fx(d)3111 4782 y Fv(1)p
3068 4802 123 4 v 3079 4893 a FA(2)p Fy(d)634 5119 y
Fw(R)718 5134 y Fx(";N)838 5119 y FA(\()p Fy(q)919 5134
y Fv(1)958 5119 y FA(\))28 b(:=)1143 4983 y Fz(Z)1199
5209 y Fp(T)1243 5190 y Fq(2)1298 5119 y Fy(F)14 b FA(\()p
Fy(')1477 5134 y Fv(1)1516 5119 y Fy(;)j(q)1603 5134
y Fv(1)1643 5119 y Fy(;)g(\016)31 b FA(=)d(0\))21 b Fw(\000)i
Fy(F)14 b FA(\()p Fy(')2252 5134 y Fv(1)2291 5119 y Fy(;)j(q)2378
5134 y Fv(1)2440 5119 y FA(+)22 b Fy(q)2581 5134 y Fv(2)2620
5119 y FA(\()p Fy(q)2701 5134 y Fv(1)2741 5119 y FA(\))g(+)g
Fy(p)p FA(\()p Fy(q)3029 5134 y Fv(1)3069 5119 y FA(\))p
Fy(;)17 b(\016)t FA(\))1149 5378 y(+)1257 5311 y(1)p
1257 5355 49 4 v 1257 5446 a(2)1316 5378 y Fy(f)11 b
FA(\()p Fy(')1477 5393 y Fv(1)1516 5378 y Fy(;)17 b(q)1603
5393 y Fv(1)1664 5378 y FA(+)22 b Fy(q)1805 5393 y Fv(2)1845
5378 y FA(\()p Fy(q)1926 5393 y Fv(1)1965 5378 y FA(\))h(+)f
Fy(p)p FA(\()p Fy(q)2254 5393 y Fv(1)2293 5378 y FA(\))p
Fy(;)17 b(\016)t FA(\)\()p Fy(q)2541 5393 y Fv(2)2580
5378 y FA(\()p Fy(q)2661 5393 y Fv(1)2701 5378 y FA(\))22
b(+)g Fy(p)p FA(\()p Fy(q)2989 5393 y Fv(1)3029 5378
y FA(\)\))p eop end
%%Page: 10 10
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b Fy(C)1533 281 y Fv(1)1573 266 y FA(\()p Fy(R)q FA(\))27
b Fw(\025)h Fy(C)1926 281 y Fv(0)1966 266 y FA(\()p Fy(R)q
FA(\))p Fo(,)698 493 y Fw(jR)810 508 y Fx(";N)930 493
y FA(\()p Fy(q)1011 508 y Fv(1)1051 493 y FA(\))p Fw(j)82
b(\024)i Fy(C)1430 508 y Fv(1)1469 493 y FA(\()p Fy(R)q
FA(\))1620 383 y Fz(\020)1679 493 y Fy(\016)26 b FA(+)c
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Fy(N)2316 533 y Fv(2)2366 383 y Fz(\021)100 493 y FA(\(2.14\))577
616 y Fz(\014)577 676 y(\014)577 735 y(\014)610 730 y
Fw(R)694 689 y Fu(0)694 755 y Fx(";N)814 730 y FA(\()p
Fy(q)895 745 y Fv(1)935 730 y FA(\)[)p Fy(h)p FA(])1083
616 y Fz(\014)1083 676 y(\014)1083 735 y(\014)1199 730
y Fw(\024)84 b Fy(C)1430 745 y Fv(1)1469 730 y FA(\()p
Fy(R)q FA(\))1620 620 y Fz(\020)1679 730 y Fy(\016)26
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Fv(1)2120 730 y FA(+)2267 663 y(1)p 2228 708 V 2228 799
a Fy(N)2316 770 y Fv(2)2366 620 y Fz(\021)2425 730 y
Fw(j)p Fy(h)p Fw(j)2537 747 y Fx(H)2600 728 y Fq(1)2639
730 y Fy(;)216 b Fw(8)p Fy(h)28 b Fw(2)g Fy(Q)3192 745
y Fv(1)3248 730 y Fy(:)-3175 b FA(\(2.15\))100 947 y
Fo(Pr)-5 b(o)g(of.)41 b FA(In)33 b(the)g(App)s(endix.)2547
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1372 y(By)41 b(\(2.13\),)h(this)f(is)g(equiv)-5 b(alen)m(t)43
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y FA(\()p Fy(q)1670 1752 y Fv(1)1710 1737 y FA(\))27
b Fw(\021)1880 1627 y Fz(\020)1940 1737 y Fw(A)p FA(\()p
Fy(q)2101 1752 y Fv(1)2140 1737 y FA(\))22 b Fw(\000)2300
1602 y Fz(Z)2355 1827 y Fp(T)2399 1808 y Fq(2)2455 1737
y Fy(a)2506 1752 y Fv(2)p Fx(d)p Fu(\000)p Fv(1)2672
1737 y FA(\()p Fy(')2774 1752 y Fv(1)2813 1737 y FA(\))2861
1670 y Fy(q)2908 1634 y Fv(2)p Fx(d)2904 1694 y Fv(1)p
2861 1714 123 4 v 2873 1806 a FA(2)p Fy(d)2994 1627 y
Fz(\021)3075 1737 y FA(+)g Fw(R)3257 1752 y Fx(";N)3377
1737 y FA(\()p Fy(q)3458 1752 y Fv(1)3498 1737 y FA(\))100
1967 y(where)33 b(the)g(quadratic)h(form)1045 2200 y
Fw(A)p FA(\()p Fy(q)t FA(\))27 b(:=)1406 2064 y Fz(Z)1461
2290 y Fp(T)1505 2271 y Fq(2)1570 2132 y FA(\(2)22 b(+)g
Fy(")p FA(\))p 1570 2177 291 4 v 1691 2268 a(2)1871 2200
y(\()p Fy(@)1960 2215 y Fx(')2006 2224 y Fq(2)2045 2200
y Fy(q)t FA(\))2130 2159 y Fv(2)2191 2200 y FA(+)g(\()p
Fy(@)2378 2215 y Fx(')2424 2224 y Fq(1)2464 2200 y Fy(q)t
FA(\)\()p Fy(@)2638 2215 y Fx(')2684 2224 y Fq(2)2722
2200 y Fy(q)t FA(\))100 2430 y(is)36 b(p)s(ositiv)m(e)h(de\014nite)f
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y FA(and)g(zero-de\014nite)h(on)e Fy(Q)3264 2445 y Fv(0)3304
2430 y FA(.)53 b(F)-8 b(or)34 b Fy(q)3604 2445 y Fv(1)3677
2430 y FA(=)100 2546 y Fy(q)143 2561 y Fv(+)224 2546
y FA(+)22 b Fy(q)365 2561 y Fv(0)427 2546 y FA(+)g Fy(q)568
2561 y Fu(\000)655 2546 y Fw(2)28 b Fy(Q)826 2561 y Fv(1)866
2546 y FA(,)100 2749 y(\(2.17\))264 b Fw(A)p FA(\()p
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FA(\()p Fy(q)1145 2764 y Fv(+)1225 2749 y FA(+)22 b Fy(q)1366
2764 y Fv(0)1428 2749 y FA(+)g Fy(q)1569 2764 y Fu(\000)1629
2749 y FA(\))27 b(=)h Fw(A)p FA(\()p Fy(q)1959 2764 y
Fv(+)2018 2749 y FA(\))22 b(+)g Fw(A)p FA(\()p Fy(q)2337
2764 y Fu(\000)2395 2749 y FA(\))28 b(=)2575 2682 y Fy(\013)2637
2697 y Fv(+)p 2575 2726 122 4 v 2611 2818 a FA(2)2706
2749 y Fw(j)p Fy(q)2777 2764 y Fv(+)2836 2749 y Fw(j)2864
2708 y Fv(2)2864 2775 y Fx(H)2927 2756 y Fq(1)2987 2749
y Fw(\000)3097 2682 y Fy(\013)3159 2697 y Fu(\000)p 3097
2726 V 3133 2818 a FA(2)3228 2749 y Fw(j)p Fy(q)3299
2764 y Fu(\000)3358 2749 y Fw(j)3386 2708 y Fv(2)3386
2775 y Fx(H)3449 2756 y Fq(1)100 2958 y FA(for)f(suitable)i(p)s(ositiv)
m(e)g(constan)m(ts)g Fy(\013)1448 2973 y Fv(+)1507 2958
y FA(,)g Fy(\013)1625 2973 y Fu(\000)1684 2958 y FA(,)g(b)s(ounded)f(a)
m(w)m(a)m(y)h(from)f(0)f(b)m(y)i(constan)m(ts)g(indep)s(enden)m(t)100
3074 y(of)j Fy(")p FA(.)199 3190 y(W)-8 b(e)33 b(shall)g(pro)m(v)m(e)h
(the)f(existence)i(of)d(critical)i(p)s(oin)m(ts)f(of)f(\010)2352
3205 y Fx(";N)2504 3190 y FA(in)h Fy(B)2692 3205 y Fv(2)p
Fx(R)2818 3190 y FA(of)f(\\linking)h(t)m(yp)s(e".)100
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Fx(";N)3344 3471 y FG(.)49 b FA(W)-8 b(e)47 b(can)100
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(space)f Fy(Q)2780 3719 y Fv(1)2820 3704 y FA(.)100 3873
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y Fv(1)3599 3873 y Fy(;)17 b Ft(R)p FA(\))100 3989 y(as)1329
4102 y Fz(e)1322 4127 y FA(\010)1392 4142 y Fx(";N)1512
4127 y FA(\()p Fy(q)1593 4142 y Fv(1)1632 4127 y FA(\))28
b(:=)f(\000\()p Fy(q)1970 4142 y Fv(1)2010 4127 y FA(\))22
b(+)2191 4102 y Fz(e)2168 4127 y Fw(R)2252 4142 y Fx(";N)2372
4127 y FA(\()p Fy(q)2453 4142 y Fv(1)2493 4127 y FA(\))100
4303 y(where)404 4278 y Fz(e)381 4303 y Fw(R)465 4318
y Fx(";N)613 4303 y FA(:)28 b Fy(Q)745 4318 y Fv(1)812
4303 y Fw(!)f Ft(R)33 b FA(is)1292 4529 y Fz(e)1269 4554
y Fw(R)1353 4569 y Fx(";N)1473 4554 y FA(\()p Fy(q)1554
4569 y Fv(1)1594 4554 y FA(\))27 b(:=)h Fy(\025)1847
4443 y Fz(\020)1916 4481 y Fw(j)p Fy(q)1987 4496 y Fv(1)2027
4481 y Fw(j)2055 4444 y Fv(2)2055 4506 y Fx(H)2113 4515
y Fq(1)p 1916 4531 235 4 v 1976 4622 a Fy(R)2051 4593
y Fv(2)2161 4443 y Fz(\021)2221 4554 y Fw(R)2305 4569
y Fx(";N)2424 4554 y FA(\()p Fy(q)2505 4569 y Fv(1)2545
4554 y FA(\))100 4768 y(and)k Fy(\025)c FA(:)g([0)p Fy(;)17
b FA(+)p Fw(1)p FA(\))26 b Fw(!)i FA([0)p Fy(;)17 b FA(1])32
b(is)h(a)f(smo)s(oth,)h(non-increasing,)g(cut-o\013)g(function)g(suc)m
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281 y Fv(+)1576 266 y Fw(j)1604 282 y Fx(H)1667 263 y
Fq(1)1734 266 y FA(=)27 b Fy(\032)h(<)g(R)33 b FA(w)m(e)h(ha)m(v)m(e)
484 484 y Fz(e)476 509 y FA(\010)546 524 y Fx(";N)666
509 y FA(\()p Fy(q)747 524 y Fv(+)806 509 y FA(\))28
b(=)g(\010)1046 524 y Fx(";N)1165 509 y FA(\()p Fy(q)1246
524 y Fv(+)1306 509 y FA(\))83 b(=)h Fw(A)p FA(\()p Fy(q)1748
524 y Fv(+)1807 509 y FA(\))22 b Fw(\000)1967 373 y Fz(Z)2022
599 y Fp(T)2066 580 y Fq(2)2122 509 y Fy(a)2173 524 y
Fv(2)p Fx(d)p Fu(\000)p Fv(1)2339 509 y FA(\()p Fy(')2441
524 y Fv(1)2480 509 y FA(\))2528 439 y Fy(q)2575 402
y Fv(2)p Fx(d)2571 463 y Fv(+)p 2528 486 123 4 v 2539
577 a FA(2)p Fy(d)2683 509 y FA(+)g Fw(R)2865 524 y Fx(";N)2984
509 y FA(\()p Fy(q)3065 524 y Fv(+)3125 509 y FA(\))1427
768 y Fw(\025)1597 700 y Fy(\013)1659 715 y Fv(+)p 1597
745 122 4 v 1633 836 a FA(2)1728 768 y Fy(\032)1778 727
y Fv(2)1840 768 y Fw(\000)h Fy(\024)1996 783 y Fv(1)2035
768 y Fy(\032)2085 727 y Fv(2)p Fx(d)2184 768 y Fw(\000)2283
657 y Fz(\020)2343 768 y Fy(\016)j FA(+)c Fw(j)p Fy(")p
Fw(j)p Fy(\015)2668 727 y Fu(\000)p Fv(1)2783 768 y FA(+)2931
700 y(1)p 2891 745 128 4 v 2891 836 a Fy(N)2979 807 y
Fv(2)3029 657 y Fz(\021)3089 768 y Fy(C)3159 783 y Fv(2)3198
768 y FA(\()p Fy(R)q FA(\))p Fy(:)-3276 b FA(\(2.20\))100
984 y(No)m(w)33 b(w)m(e)i(\014x)e Fy(\032)c(>)g FA(0)k(small)h(suc)m(h)
g(that)f(\()p Fy(\013)1659 999 y Fv(+)1718 984 y Fy(\032)1768
948 y Fv(2)1808 984 y Fy(=)p FA(2\))22 b Fw(\000)h Fy(\024)2122
999 y Fv(1)2162 984 y Fy(\032)2212 948 y Fv(2)p Fx(d)2317
984 y Fw(\025)29 b Fy(\013)2485 999 y Fv(+)2544 984 y
Fy(\032)2594 948 y Fv(2)2634 984 y Fy(=)p FA(4.)45 b(Next,)34
b(for)f(\()p Fy(\016)26 b FA(+)c Fw(j)p Fy(")p Fw(j)p
Fy(\015)3583 948 y Fu(\000)p Fv(1)3677 984 y FA(+)100
1100 y Fy(N)188 1064 y Fu(\000)p Fv(2)282 1100 y FA(\))p
Fy(C)390 1115 y Fv(2)430 1100 y FA(\()p Fy(R)q FA(\))27
b Fw(\024)33 b Fy(\013)780 1115 y Fv(+)839 1100 y Fy(\032)889
1064 y Fv(2)929 1100 y Fy(=)p FA(8)f(w)m(e)i(get)e(b)m(y)i(\(2.20\))518
1276 y Fz(e)510 1301 y FA(\010)580 1316 y Fx(";N)700
1301 y FA(\()p Fy(q)781 1316 y Fv(+)840 1301 y FA(\))28
b Fw(\025)1021 1234 y Fy(\013)1083 1249 y Fv(+)p 1021
1278 122 4 v 1057 1370 a FA(8)1152 1301 y Fy(\032)1202
1260 y Fv(2)1269 1301 y FA(=:)g Fy(!)j(>)c FA(0)p Fy(;)212
b Fw(8)33 b Fy(q)2014 1316 y Fv(+)2101 1301 y Fw(2)28
b Fy(Q)2272 1316 y Fv(1)2334 1301 y Fw(\\)22 b Fy(Q)2499
1260 y Fv(+)2656 1301 y FA(with)98 b Fw(j)p Fy(q)3014
1316 y Fv(+)3073 1301 y Fw(j)28 b FA(=)f Fy(\032)33 b(:)100
1510 y FG(\(ii\))f FA(Let)100 1715 y Fy(B)174 1730 y
Fv(1)296 1715 y FA(:=)482 1604 y Fz(n)549 1715 y Fy(q)592
1730 y Fv(1)659 1715 y FA(=)c Fy(q)806 1730 y Fv(0)867
1715 y FA(+)22 b Fy(q)1008 1730 y Fu(\000)1090 1715 y
FA(+)g Fy(r)1232 1730 y Fv(2)1271 1715 y Fy(e)1316 1730
y Fv(+)1408 1715 y FA(with)66 b Fw(j)p Fy(q)1734 1730
y Fv(0)1795 1715 y FA(+)22 b Fy(q)1936 1730 y Fu(\000)1995
1715 y Fw(j)2023 1732 y Fx(H)2086 1713 y Fq(1)2153 1715
y Fw(\024)28 b Fy(r)2302 1730 y Fv(1)2341 1715 y Fy(;)17
b(q)2428 1730 y Fu(\000)2515 1715 y Fw(2)28 b Fy(Q)2686
1730 y Fv(1)2748 1715 y Fw(\\)22 b Fy(Q)2913 1730 y Fu(\000)2973
1604 y Fz(o)3067 1715 y Fw(\032)28 b Fy(@)5 b(W)3334
1674 y Fu(\000)100 1932 y Fy(B)174 1947 y Fv(2)296 1932
y FA(:=)482 1821 y Fz(n)549 1932 y Fy(q)592 1947 y Fv(1)659
1932 y FA(=)28 b Fy(q)806 1947 y Fv(0)867 1932 y FA(+)22
b Fy(q)1008 1947 y Fu(\000)1090 1932 y FA(+)g Fy(r)s(e)1280
1947 y Fv(+)1371 1932 y FA(with)66 b Fw(j)p Fy(q)1697
1947 y Fv(0)1758 1932 y FA(+)22 b Fy(q)1899 1947 y Fu(\000)1959
1932 y Fw(j)1987 1948 y Fx(H)2050 1929 y Fq(1)2116 1932
y FA(=)27 b Fy(r)2263 1947 y Fv(1)2303 1932 y Fy(;)17
b(q)2390 1947 y Fu(\000)2477 1932 y Fw(2)28 b Fy(Q)2648
1947 y Fv(1)2709 1932 y Fw(\\)23 b Fy(Q)2875 1947 y Fu(\000)2934
1932 y Fy(;)50 b(r)30 b Fw(2)e FA([0)p Fy(;)17 b(r)3343
1947 y Fv(2)3382 1932 y FA(])3409 1821 y Fz(o)3503 1932
y Fw(\032)28 b Fy(@)5 b(W)3770 1891 y Fu(\000)100 2131
y FA(and)32 b(c)m(ho)s(ose)i Fy(r)643 2146 y Fv(1)682
2131 y Fy(;)17 b(r)770 2146 y Fv(2)837 2131 y Fy(>)27
b FA(2)p Fy(R)q FA(.)44 b(F)-8 b(or)32 b Fy(q)1353 2146
y Fv(1)1420 2131 y FA(=)27 b Fy(q)1566 2146 y Fv(0)1628
2131 y FA(+)22 b Fy(q)1769 2146 y Fu(\000)1851 2131 y
FA(+)g Fy(r)s(e)2041 2146 y Fv(+)2127 2131 y Fw(2)28
b Fy(B)2295 2146 y Fv(1)2357 2131 y Fw([)23 b Fy(B)2520
2146 y Fv(2)280 2329 y Fz(e)272 2354 y FA(\010)342 2369
y Fx(";N)462 2354 y FA(\()p Fy(q)543 2369 y Fv(1)583
2354 y FA(\))28 b(=)f(\000\()p Fy(q)894 2369 y Fv(1)934
2354 y FA(\))83 b(=)h Fw(A)p FA(\()p Fy(q)1376 2369 y
Fv(1)1415 2354 y FA(\))22 b Fw(\000)1575 2219 y Fz(Z)1630
2444 y Fp(T)1674 2425 y Fq(2)1730 2354 y Fy(a)1781 2369
y Fv(2)p Fx(d)p Fu(\000)p Fv(1)1947 2354 y FA(\()p Fy(')2049
2369 y Fv(1)2088 2354 y FA(\)\()p Fy(q)2207 2369 y Fv(0)2269
2354 y FA(+)g Fy(q)2410 2369 y Fu(\000)2491 2354 y FA(+)h
Fy(r)s(e)2682 2369 y Fv(+)2740 2354 y FA(\))2778 2313
y Fv(2)p Fx(d)1055 2631 y FA(=)84 b Fw(\000)1302 2564
y Fy(\013)1364 2579 y Fu(\000)p 1302 2608 V 1339 2700
a FA(2)1434 2631 y Fw(j)p Fy(q)1505 2646 y Fu(\000)1564
2631 y Fw(j)1592 2590 y Fv(2)1592 2657 y Fx(H)1655 2638
y Fq(1)1715 2631 y FA(+)22 b Fy(r)1860 2590 y Fv(2)1899
2631 y Fw(A)p FA(\()p Fy(e)2062 2646 y Fv(+)2121 2631
y FA(\))g Fw(\000)2281 2496 y Fz(Z)2336 2721 y Fp(T)2380
2702 y Fq(2)2436 2631 y Fy(a)2487 2646 y Fv(2)p Fx(d)p
Fu(\000)p Fv(1)2653 2631 y FA(\()p Fy(')2755 2646 y Fv(1)2794
2631 y FA(\))2842 2564 y(\()p Fy(q)2923 2579 y Fv(0)2985
2564 y FA(+)g Fy(q)3126 2579 y Fu(\000)3207 2564 y FA(+)g
Fy(r)s(e)3397 2579 y Fv(+)3456 2564 y FA(\))3494 2528
y Fv(2)p Fx(d)p 2842 2608 728 4 v 3156 2700 a FA(2)p
Fy(d)1055 2894 y Fw(\024)83 b(\000)1302 2827 y Fy(\013)1364
2842 y Fu(\000)p 1302 2871 122 4 v 1339 2963 a FA(2)1434
2894 y Fw(j)p Fy(q)1505 2909 y Fu(\000)1564 2894 y Fw(j)1592
2853 y Fv(2)1592 2920 y Fx(H)1655 2901 y Fq(1)1715 2894
y FA(+)22 b Fy(r)1860 2853 y Fv(2)1899 2894 y Fw(A)p
FA(\()p Fy(e)2062 2909 y Fv(+)2121 2894 y FA(\))g Fw(\000)h
Fy(\013)2360 2759 y Fz(Z)2415 2984 y Fp(T)2459 2965 y
Fq(2)2498 2894 y FA(\()p Fy(q)2579 2909 y Fv(0)2641 2894
y FA(+)f Fy(q)2782 2909 y Fu(\000)2864 2894 y FA(+)g
Fy(r)s(e)3054 2909 y Fv(+)3113 2894 y FA(\))3151 2853
y Fv(2)p Fx(d)100 2894 y FA(\(2.21\))100 3125 y(b)s(ecause)34
b Fy(a)512 3140 y Fv(2)p Fx(d)p Fu(\000)p Fv(1)678 3125
y FA(\()p Fy(')780 3140 y Fv(1)819 3125 y FA(\))p Fy(=)p
FA(2)p Fy(d)27 b Fw(\025)h Fy(\013)g(>)g FA(0.)43 b(No)m(w,)33
b(b)m(y)h(H\177)-49 b(older)33 b(inequalit)m(y)h(and)f(orthogonalit)m
(y)784 3215 y Fz(Z)840 3441 y Fp(T)884 3422 y Fq(2)923
3351 y FA(\()p Fy(q)1004 3366 y Fv(0)1065 3351 y FA(+)22
b Fy(q)1206 3366 y Fu(\000)1288 3351 y FA(+)g Fy(r)s(e)1478
3366 y Fv(+)1537 3351 y FA(\))1575 3310 y Fv(2)p Fx(d)1734
3351 y Fw(\025)83 b Fy(\024)1950 3366 y Fv(2)1990 3240
y Fz(\020)2066 3215 y(Z)2121 3441 y Fp(T)2165 3422 y
Fq(2)2204 3351 y FA(\()p Fy(q)2285 3366 y Fv(0)2347 3351
y FA(+)22 b Fy(q)2488 3366 y Fu(\000)2569 3351 y FA(+)g
Fy(r)s(e)2759 3366 y Fv(+)2818 3351 y FA(\))2856 3310
y Fv(2)2896 3240 y Fz(\021)2955 3263 y Fx(d)1734 3614
y FA(=)84 b Fy(\024)1950 3629 y Fv(2)1990 3503 y Fz(\020)2066
3478 y(Z)2121 3704 y Fp(T)2165 3685 y Fq(2)2221 3614
y Fy(q)2268 3573 y Fv(2)2264 3638 y(0)2329 3614 y FA(+)22
b Fy(q)2474 3573 y Fv(2)2470 3638 y Fu(\000)2552 3614
y FA(+)g Fy(r)2697 3573 y Fv(2)2736 3614 y Fy(e)2781
3573 y Fv(2)2781 3638 y(+)2840 3503 y Fz(\021)2899 3525
y Fx(d)1734 3828 y Fw(\025)83 b Fy(\024)1950 3843 y Fv(3)1990
3828 y FA(\()p Fy(q)2075 3787 y Fv(2)2071 3853 y(0)2136
3828 y FA(+)22 b Fy(r)2281 3787 y Fv(2)2320 3828 y FA(\))2358
3787 y Fx(d)2426 3828 y Fw(\025)28 b Fy(\024)2587 3843
y Fv(3)2627 3828 y FA(\()p Fy(q)2712 3787 y Fv(2)p Fx(d)2708
3853 y Fv(0)2810 3828 y FA(+)22 b Fy(r)2955 3787 y Fv(2)p
Fx(d)3030 3828 y FA(\))100 4002 y(and)32 b(b)m(y)i(\(2.21\))e(w)m(e)h
(deduce)618 4182 y Fz(e)611 4207 y FA(\010)681 4222 y
Fx(";N)801 4207 y FA(\()p Fy(q)882 4222 y Fv(0)943 4207
y FA(+)22 b Fy(q)1084 4222 y Fu(\000)1166 4207 y FA(+)g
Fy(r)s(e)1356 4222 y Fv(+)1415 4207 y FA(\))27 b Fw(\024)h
FA(\()p Fy(\024)1679 4222 y Fv(+)1739 4207 y Fy(r)1786
4166 y Fv(2)1847 4207 y Fw(\000)22 b Fy(\024)2002 4222
y Fv(3)2042 4207 y Fy(r)2089 4166 y Fv(2)p Fx(d)2164
4207 y FA(\))h Fw(\000)2324 4096 y Fz(\020)2394 4140
y Fy(\013)2456 4155 y Fu(\000)p 2394 4184 V 2430 4275
a FA(2)2525 4207 y Fw(j)p Fy(q)2596 4222 y Fu(\000)2655
4207 y Fw(j)2683 4166 y Fv(2)2683 4233 y Fx(H)2746 4214
y Fq(1)2806 4207 y FA(+)f Fy(\024)2960 4222 y Fv(3)3000
4207 y Fy(q)3047 4166 y Fv(2)p Fx(d)3043 4232 y Fv(0)3122
4096 y Fz(\021)3214 4207 y Fy(:)100 4423 y FA(No)m(w)33
b(w)m(e)g(\014x)h Fy(r)648 4438 y Fv(2)720 4423 y FA(large)e(suc)m(h)i
(that)e Fy(\024)1445 4438 y Fv(+)1505 4423 y Fy(r)1552
4387 y Fv(2)1549 4448 y(2)1613 4423 y Fw(\000)22 b Fy(\024)1768
4438 y Fv(3)1808 4423 y Fy(r)1855 4387 y Fv(2)p Fx(d)1852
4448 y Fv(2)1958 4423 y Fw(\024)28 b FA(0)33 b(and)f(therefore)1018
4585 y Fz(e)1010 4611 y FA(\010)1080 4626 y Fx(";N)1200
4611 y FA(\()p Fy(q)1281 4626 y Fv(1)1321 4611 y FA(\))27
b Fw(\024)h Fy(\024)1547 4626 y Fv(+)1607 4611 y Fy(r)1654
4570 y Fv(2)1651 4635 y(2)1715 4611 y Fw(\000)22 b Fy(\024)1870
4626 y Fv(3)1910 4611 y Fy(r)1957 4570 y Fv(2)p Fx(d)1954
4635 y Fv(2)2060 4611 y Fw(\024)28 b FA(0)195 b Fw(8)p
Fy(q)2507 4626 y Fv(1)2575 4611 y Fw(2)28 b Fy(B)2743
4626 y Fv(1)2815 4611 y Fy(:)100 4786 y FA(Next,)33 b(setting)g
Fy(M)39 b FA(:=)28 b(max)1130 4802 y Fx(r)r Fu(2)p Fv([0)p
Fx(;r)1318 4811 y Fq(2)1352 4802 y Fv(])1376 4786 y FA(\()p
Fy(\024)1470 4801 y Fv(+)1529 4786 y Fy(r)1576 4750 y
Fv(2)1637 4786 y Fw(\000)23 b Fy(\024)1793 4801 y Fv(3)1832
4786 y Fy(r)1879 4750 y Fv(2)p Fx(d)1955 4786 y FA(\),)32
b(w)m(e)i(\014x)f Fy(r)2378 4801 y Fv(1)2450 4786 y FA(large)f(suc)m(h)
i(that)1019 4927 y Fy(\013)1081 4942 y Fu(\000)p 1019
4972 V 1055 5063 a FA(2)1150 4994 y Fw(j)p Fy(q)1221
5009 y Fu(\000)1280 4994 y Fw(j)1308 4953 y Fv(2)1308
5020 y Fx(H)1371 5001 y Fq(1)1431 4994 y FA(+)23 b Fy(\024)1586
5009 y Fv(3)1625 4994 y Fy(q)1672 4953 y Fv(2)p Fx(d)1668
5019 y Fv(0)1775 4994 y Fw(\025)29 b Fy(M)205 b Fw(8)33
b(j)p Fy(q)2339 5009 y Fu(\000)2420 4994 y FA(+)22 b
Fy(q)2561 5009 y Fv(0)2601 4994 y Fw(j)27 b FA(=)h Fy(r)2804
5009 y Fv(1)100 5198 y FA(and)k(therefore)746 5353 y
Fz(e)738 5378 y FA(\010)808 5393 y Fx(";N)928 5378 y
FA(\()p Fy(q)1009 5393 y Fv(1)1049 5378 y FA(\))c Fw(\024)g
Fy(M)33 b Fw(\000)1446 5267 y Fz(\020)1516 5311 y Fy(\013)1578
5326 y Fu(\000)p 1516 5355 V 1552 5446 a FA(2)1647 5378
y Fw(j)p Fy(q)1718 5393 y Fu(\000)1777 5378 y Fw(j)1805
5337 y Fv(2)1805 5404 y Fx(H)1868 5385 y Fq(1)1928 5378
y FA(+)22 b Fy(\024)2082 5393 y Fv(3)2122 5378 y Fy(q)2169
5337 y Fv(2)p Fx(d)2165 5403 y Fv(0)2245 5267 y Fz(\021)2332
5378 y Fw(\024)28 b FA(0)195 b Fw(8)p Fy(q)2779 5393
y Fv(1)2846 5378 y Fw(2)29 b Fy(B)3015 5393 y Fv(2)3087
5378 y Fy(:)p eop end
%%Page: 13 13
TeXDict begin 13 12 bop 721 66 a Fr(QUASI-PERIODIC)30
b(SOLUTIONS)i(OF)h(F)n(OR)n(CED)g(W)-9 b(A)g(VE)33 b(EQUA)-6
b(TIONS)543 b(13)100 266 y FA(Finally)33 b(if)f Fy(q)564
281 y Fv(1)631 266 y FA(=)c Fy(q)778 281 y Fu(\000)859
266 y FA(+)22 b Fy(q)1000 281 y Fv(0)1040 266 y FA(:)834
467 y Fz(e)826 492 y FA(\010)896 507 y Fx(";N)1016 492
y FA(\()p Fy(q)1097 507 y Fv(1)1137 492 y FA(\))83 b(=)h
Fw(A)p FA(\()p Fy(q)1579 507 y Fu(\000)1638 492 y FA(\))22
b Fw(\000)1798 357 y Fz(Z)1853 582 y Fp(T)1897 563 y
Fq(2)1953 492 y Fy(a)2004 507 y Fv(2)p Fx(d)p Fu(\000)p
Fv(1)2170 492 y FA(\()p Fy(')2272 507 y Fv(1)2311 492
y FA(\))2359 425 y Fy(q)2406 389 y Fv(2)p Fx(d)2402 450
y Fv(1)p 2359 469 123 4 v 2370 561 a FA(2)p Fy(d)2513
492 y FA(+)2634 467 y Fz(e)2611 492 y Fw(R)2695 507 y
Fx(";N)2815 492 y FA(\()p Fy(q)2896 507 y Fv(1)2936 492
y FA(\))1258 717 y Fw(\024)83 b(j)1469 692 y Fz(e)1446
717 y Fw(R)1530 732 y Fx(";N)1650 717 y FA(\()p Fy(q)1731
732 y Fv(1)1770 717 y FA(\))p Fw(j)28 b(\024)g Fy(C)2039
732 y Fv(2)2078 717 y FA(\()p Fy(R)q FA(\)\()p Fy(\016)e
FA(+)c Fw(j)p Fy(")p Fw(j)p Fy(\015)2592 676 y Fu(\000)p
Fv(1)2707 717 y FA(+)g Fy(N)2893 676 y Fu(\000)p Fv(2)2988
717 y FA(\))-2926 b(\(2.22\))100 897 y(and)32 b(so)417
872 y Fz(e)409 897 y FA(\010)479 912 y Fx(";N)599 897
y FA(\()p Fy(q)680 912 y Fv(1)720 897 y FA(\))27 b Fw(\024)h
Fy(!)t(=)p FA(2)64 b(if)h Fy(C)1309 912 y Fv(2)1348 897
y FA(\()p Fy(R)q FA(\)\()p Fy(\016)26 b FA(+)c Fw(j)p
Fy(")p Fw(j)p Fy(\015)1862 861 y Fu(\000)p Fv(1)1978
897 y FA(+)g Fy(N)2164 861 y Fu(\000)p Fv(2)2259 897
y FA(\))27 b Fw(\024)h Fy(!)t(=)p FA(2.)1056 b Fd(\003)199
1117 y FA(W)-8 b(e)33 b(in)m(tro)s(duce)h(the)f(minimax)h(class)961
1287 y Fw(S)i FA(:=)1187 1176 y Fz(n)1254 1287 y Fy( )31
b Fw(2)d Fy(C)7 b FA(\()p 1557 1206 165 4 v Fy(W)1663
1258 y Fu(\000)1722 1287 y Fy(;)17 b(Q)p FA(\))65 b Fw(j)g
Fy( )31 b FA(=)d(Id)65 b(on)g Fy(@)5 b(W)2721 1246 y
Fu(\000)2781 1176 y Fz(o)2864 1287 y Fy(:)100 1476 y
FA(The)41 b(maps)g(of)e Fw(S)48 b FA(ha)m(v)m(e)42 b(an)e(imp)s(ortan)m
(t)g(in)m(tersection)i(prop)s(ert)m(y)-8 b(,)43 b(see)e(e.g.)67
b(Prop)s(osition)40 b(5.9)g(of)100 1592 y([R3)o(].)100
1770 y FG(Prop)s(osition)d(1.)42 b(\()p Fy(S)944 1734
y Fv(+)1040 1770 y FG(and)c Fy(W)1362 1734 y Fu(\000)1458
1770 y FG(link)g(with)f(resp)s(ect)h(to)f Fw(S)7 b FG(.\))1207
1933 y Fy( )32 b Fw(2)c(S)135 b FA(=)-17 b Fw(\))127
b Fy( )t FA(\()p Fy(W)2088 1892 y Fu(\000)2147 1933 y
FA(\))22 b Fw(\\)g Fy(S)2361 1892 y Fv(+)2448 1933 y
Fw(6)p FA(=)28 b Fw(;)17 b Fy(:)199 2111 y FA(De\014ne)33
b(the)g(minimax)h(linking)g(lev)m(el)1289 2283 y Fw(K)1365
2298 y Fx(";N)1512 2283 y FA(:=)40 b(inf)1643 2346 y
Fx( )r Fu(2S)1816 2283 y FA(max)1802 2350 y Fx(q)r Fu(2)p
Fx(W)1960 2331 y Fm(\000)2036 2258 y Fz(e)2028 2283 y
FA(\010)2098 2298 y Fx(";N)2218 2283 y FA(\()p Fy( )t
FA(\()p Fy(q)2404 2298 y Fv(1)2444 2283 y FA(\)\))17
b Fy(:)100 2503 y FA(By)33 b(the)g(in)m(tersection)h(prop)s(ert)m(y)g
(of)e(Prop)s(osition)h(1)f(and)g(Lemma)i(2.5-\()p Fy(i)p
FA(\))785 2680 y(max)755 2747 y Fx(q)787 2756 y Fq(1)821
2747 y Fu(2)p Fx(W)945 2729 y Fm(\000)1021 2655 y Fz(e)1013
2680 y FA(\010)1083 2695 y Fx(";N)1203 2680 y FA(\()p
Fy( )t FA(\()p Fy(q)1389 2695 y Fv(1)1429 2680 y FA(\)\))27
b Fw(\025)52 b FA(min)1637 2747 y Fx(q)1669 2756 y Fq(1)1704
2747 y Fu(2)p Fx(S)1798 2729 y Fq(+)1872 2655 y Fz(e)1865
2680 y FA(\010)1935 2695 y Fx(";N)2054 2680 y FA(\()p
Fy(q)2135 2695 y Fv(1)2175 2680 y FA(\))28 b Fw(\025)g
Fy(!)j(>)c FA(0)195 b Fw(8)p Fy( )32 b Fw(2)c(S)100 2895
y FA(and)k(therefore)1619 3017 y Fw(K)1695 3032 y Fx(";N)1842
3017 y Fy(>)27 b(!)k(>)d FA(0)17 b Fy(:)100 3160 y FA(Moreo)m(v)m(er,)
34 b(since)g(Id)f Fw(2)28 b(S)40 b FA(and)33 b(\(2.18\))451
3354 y Fw(K)527 3369 y Fx(";N)729 3354 y Fw(\024)114
b FA(max)890 3422 y Fx(q)922 3431 y Fq(1)956 3422 y Fu(2)p
Fx(W)1080 3403 y Fm(\000)1156 3329 y Fz(e)1148 3354 y
FA(\010)1218 3369 y Fx(";N)1338 3354 y FA(\()p Fy(q)1419
3369 y Fv(1)1459 3354 y FA(\))27 b Fw(\024)58 b FA(max)1629
3422 y Fx(q)1661 3431 y Fq(1)1696 3422 y Fu(2)p Fx(W)1820
3403 y Fm(\000)1888 3244 y Fz(\020)1947 3354 y FA(\000\()p
Fy(q)2089 3369 y Fv(1)2129 3354 y FA(\))22 b(+)2310 3329
y Fz(e)2287 3354 y Fw(R)2371 3369 y Fx(";N)2491 3354
y FA(\()p Fy(q)2572 3369 y Fv(1)2611 3354 y FA(\))2649
3244 y Fz(\021)729 3618 y Fw(\024)114 b FA(max)890 3685
y Fx(q)922 3694 y Fq(1)956 3685 y Fu(2)p Fx(W)1080 3666
y Fm(\000)1148 3507 y Fz(\020)1218 3550 y Fy(\013)1280
3565 y Fv(+)p 1218 3595 122 4 v 1254 3686 a FA(2)1349
3618 y Fw(j)p Fy(q)1420 3633 y Fv(+)1479 3618 y Fw(j)1507
3576 y Fv(2)1507 3643 y Fx(H)1570 3624 y Fq(1)1630 3618
y FA(+)1738 3550 y Fy(\013)1800 3565 y Fu(\000)p 1738
3595 V 1774 3686 a FA(2)1869 3618 y Fw(j)p Fy(q)1940
3633 y Fu(\000)1999 3618 y Fw(j)2027 3576 y Fv(2)2027
3643 y Fx(H)2090 3624 y Fq(1)2151 3618 y FA(+)2249 3482
y Fz(Z)2304 3707 y Fp(T)2348 3689 y Fq(2)2404 3618 y
Fy(\024q)2507 3576 y Fv(2)p Fx(d)2503 3642 y Fv(1)2583
3507 y Fz(\021)2664 3618 y FA(+)22 b(1)28 b Fw(\024)g(K)3020
3633 y Fu(1)3122 3618 y Fy(<)g FA(+)p Fw(1)-3302 b FA(\(2.23\))100
3839 y(where)37 b Fw(K)461 3854 y Fu(1)571 3839 y FA(is)g(indep)s
(enden)m(t)h(of)e Fy(N)5 b(;)17 b(";)g(\015)40 b FA(since)d(the)g
(constan)m(ts)g Fy(r)2545 3854 y Fv(1)2585 3839 y Fy(;)17
b(r)2673 3854 y Fv(2)2748 3839 y FA(in)36 b(the)g(de\014nition)h(of)f
Fy(W)3694 3803 y Fu(\000)100 3955 y FA(are)c(indep)s(enden)m(t)j(of)d
Fy(N)5 b(;)17 b(";)g(\015)5 b FA(.)199 4072 y(W)-8 b(e)29
b(deduce,)i(b)m(y)e(the)f(linking)i(theorem)f(the)f(existence)j(of)d(a)
g(\(P)m(alais-Smale\))h(sequence)i(\()p Fy(q)3584 4087
y Fx(j)3621 4072 y FA(\))c Fw(2)100 4188 y Fy(Q)177 4203
y Fv(1)249 4188 y FA(at)32 b(the)h(lev)m(el)h Fw(K)834
4203 y Fx(";N)954 4188 y FA(,)e(namely)100 4368 y(\(2.24\))1154
4343 y Fz(e)1146 4368 y FA(\010)1216 4383 y Fx(";N)1336
4368 y FA(\()p Fy(q)1417 4383 y Fx(j)1454 4368 y FA(\))27
b Fw(!)h(K)1723 4383 y Fx(";N)1858 4368 y Fy(;)2105 4343
y Fz(e)2097 4368 y FA(\010)2167 4327 y Fu(0)2167 4393
y Fx(";N)2287 4368 y FA(\()p Fy(q)2368 4383 y Fx(j)2405
4368 y FA(\))f Fw(!)h FA(0)k Fy(:)100 4578 y FG(Step)46
b(3:)59 b Fc(Existence)43 b(of)c(a)h(non)m(trivial)h(critical)g(p)s
(oin)m(t.)67 b FA(Our)40 b(\014nal)h(aim)f(is)h(to)f(pro)m(v)m(e)h
(that)f(the)100 4694 y(P)m(alais-Smale)45 b(sequence)i
Fy(q)1144 4709 y Fx(j)1225 4694 y FA(con)m(v)m(erges,)j(up)45
b(to)f(subsequence,)51 b(to)44 b(some)h(non-trivial)g(critical)100
4827 y(p)s(oin)m(t)p 354 4773 47 4 v 32 w Fy(q)401 4851
y Fv(1)468 4827 y Fw(6)p FA(=)28 b(0)k(in)h(some)g(op)s(en)g(ball)g(of)
f Fy(Q)1625 4842 y Fv(1)1697 4827 y FA(where)1986 4802
y Fz(e)1979 4827 y FA(\010)2049 4842 y Fx(";N)2201 4827
y FA(and)h(\010)2461 4842 y Fx(";N)2613 4827 y FA(coincide.)100
5005 y FG(Lemma)i(2.6.)40 b Fo(Ther)-5 b(e)31 b(exists)h(a)f(c)-5
b(onstant)32 b Fy(R)1789 5020 y Fu(\003)1856 5005 y Fy(>)c
FA(0)p Fo(,)k(indep)-5 b(endent)31 b(on)g Fy(R)q Fo(-)p
Fy(")p Fo(-)p Fy(N)10 b Fo(-)p Fy(\015)5 b Fo(,)33 b(and)e(functions)
100 5122 y FA(0)42 b Fy(<)g(")355 5137 y Fv(2)394 5122
y FA(\()p Fy(R)q FA(\))h Fw(\024)g Fy(")754 5137 y Fv(1)793
5122 y FA(\()p Fy(R)q FA(\))p Fo(,)h Fy(N)1096 5137 y
Fv(2)1136 5122 y FA(\()p Fy(R)q FA(\))e Fw(\025)h Fy(N)1527
5137 y Fv(1)1567 5122 y FA(\()p Fy(R)q FA(\))g Fo(such)f(that)i(for)e
(al)5 b(l)43 b Fw(j)p Fy(")p Fw(j)p Fy(\015)2667 5085
y Fu(\000)p Fv(1)2802 5122 y Fw(\024)g Fy(")2968 5137
y Fv(2)3007 5122 y FA(\()p Fy(R)q FA(\))p Fo(,)i Fy(N)53
b Fw(\025)43 b Fy(N)3562 5137 y Fv(2)3602 5122 y FA(\()p
Fy(R)q FA(\))100 5252 y Fo(the)h(functional)745 5227
y Fz(e)738 5252 y FA(\010)808 5267 y Fx(";N)972 5252
y Fo(p)-5 b(ossesses)42 b(a)i(non-trivial)f(critic)-5
b(al)44 b(p)-5 b(oint)p 2576 5197 V 43 w Fy(q)2623 5275
y Fv(1)2707 5252 y Fw(2)45 b Fy(Q)2895 5267 y Fv(1)2979
5252 y Fo(with)e(critic)-5 b(al)44 b(value)107 5360 y
Fz(e)100 5385 y FA(\010)170 5400 y Fx(";N)290 5385 y
FA(\()p 328 5330 V Fy(q)374 5409 y Fv(1)414 5385 y FA(\))27
b(=)h Fw(K)659 5400 y Fx(";N)778 5385 y Fo(,)35 b(satisfying)f
Fw(j)p 1304 5330 V Fy(q)1351 5409 y Fv(1)1390 5385 y
Fw(j)1418 5400 y Fx(H)1476 5409 y Fq(1)1542 5385 y Fw(\024)28
b Fy(R)1721 5400 y Fu(\003)1761 5385 y Fo(.)p eop end
%%Page: 14 14
TeXDict begin 14 13 bop 100 66 a Fr(14)929 b(MASSIMILIANO)24
b(BER)-6 b(TI,)26 b(MICHELA)f(PR)n(OCESI)100 280 y Fo(Pr)-5
b(o)g(of.)41 b FA(W)-8 b(riting)763 255 y Fz(e)755 280
y FA(\010)825 295 y Fx(";N)945 280 y FA(\()p Fy(q)t FA(\))28
b(=)f(\000\()p Fy(q)t FA(\))22 b(+)1526 255 y Fz(e)1503
280 y Fw(R)1587 295 y Fx(";N)1707 280 y FA(\()p Fy(q)t
FA(\))32 b(w)m(e)i(deriv)m(e,)g(b)m(y)g(\(2.18\)-\(2.19\))107
472 y Fz(e)100 497 y FA(\010)170 512 y Fx(";N)290 497
y FA(\()p Fy(q)371 512 y Fx(j)407 497 y FA(\))22 b Fw(\000)577
430 y FA(1)p 577 474 49 4 v 577 565 a(2)643 472 y Fz(e)636
497 y FA(\010)706 456 y Fu(0)706 522 y Fx(";N)825 497
y FA(\()p Fy(q)906 512 y Fx(j)943 497 y FA(\)[)p Fy(q)1051
512 y Fx(j)1088 497 y FA(])84 b(=)g(\000\()p Fy(q)1501
512 y Fx(j)1537 497 y FA(\))22 b Fw(\000)1707 430 y FA(1)p
1707 474 V 1707 565 a(2)1766 497 y(\000)1827 456 y Fu(0)1850
497 y FA(\()p Fy(q)1931 512 y Fx(j)1968 497 y FA(\)[)p
Fy(q)2076 512 y Fx(j)2112 497 y FA(])h(+)2260 386 y Fz(\020)2342
472 y(e)2319 497 y Fw(R)2403 512 y Fx(";N)2523 497 y
FA(\()p Fy(q)2604 512 y Fx(j)2641 497 y FA(\))f Fw(\000)2810
430 y FA(1)p 2810 474 V 2810 565 a(2)2892 472 y Fz(e)2869
497 y Fw(R)2953 456 y Fu(0)2953 522 y Fx(";N)3073 497
y FA(\()p Fy(q)3154 512 y Fx(j)3191 497 y FA(\)[)p Fy(q)3299
512 y Fx(j)3336 497 y FA(])3363 386 y Fz(\021)1199 738
y FA(=)1359 628 y Fz(\020)1428 671 y FA(1)p 1428 715
V 1428 807 a(2)1509 738 y Fw(\000)1644 671 y FA(1)p 1619
715 100 4 v 1619 807 a(2)p Fy(d)1728 628 y Fz(\021)1804
603 y(Z)1859 828 y Fp(T)1903 809 y Fq(2)1959 738 y Fy(a)2010
753 y Fv(2)p Fx(d)p Fu(\000)p Fv(1)2176 738 y FA(\()p
Fy(')2278 753 y Fv(1)2317 738 y FA(\))p Fy(q)2402 697
y Fv(2)p Fx(d)2398 763 y(j)2500 738 y FA(+)2598 628 y
Fz(\020)2680 713 y(e)2658 738 y Fw(R)2742 753 y Fx(";N)2861
738 y FA(\()p Fy(q)2942 753 y Fx(j)2979 738 y FA(\))g
Fw(\000)3149 671 y FA(1)p 3149 715 49 4 v 3149 807 a(2)3230
713 y Fz(e)3208 738 y Fw(R)3292 697 y Fu(0)3292 763 y
Fx(";N)3411 738 y FA(\()p Fy(q)3492 753 y Fx(j)3529 738
y FA(\)[)p Fy(q)3637 753 y Fx(j)3674 738 y FA(])3701
628 y Fz(\021)1198 1001 y Fw(\025)84 b Fy(\013)1422 891
y Fz(\020)1491 934 y FA(1)p 1491 978 V 1491 1070 a(2)1572
1001 y Fw(\000)1707 934 y FA(1)p 1681 978 100 4 v 1681
1070 a(2)p Fy(d)1791 891 y Fz(\021)1867 866 y(Z)1922
1091 y Fp(T)1966 1072 y Fq(2)2022 1001 y Fy(q)2069 960
y Fv(2)p Fx(d)2065 1026 y(j)2166 1001 y Fw(\000)23 b
FA(\()p Fy(\016)j FA(+)c Fw(j)p Fy(")p Fw(j)p Fy(\015)2629
960 y Fu(\000)p Fv(1)2745 1001 y FA(+)g Fy(N)2931 960
y Fu(\000)p Fv(2)3025 1001 y FA(\))p Fy(C)3133 1016 y
Fv(2)3173 1001 y FA(\()p Fy(R)q FA(\))32 b Fy(:)100 1224
y FA(Therefore,)h(b)m(y)h(\(2.23\)-\(2.24\))100 1442
y(\(2.25\))703 b Fw(K)1129 1457 y Fu(1)1226 1442 y FA(+)22
b(1)g(+)g Fw(j)p Fy(q)1564 1457 y Fx(j)1600 1442 y Fw(j)1628
1457 y Fx(H)1686 1466 y Fq(1)1753 1442 y Fw(\025)28 b
Fy(\024)1914 1457 y Fv(1)1970 1306 y Fz(Z)2025 1532 y
Fp(T)2069 1513 y Fq(2)2125 1442 y Fy(q)2172 1401 y Fv(2)p
Fx(d)2168 1467 y(j)2275 1442 y FA(:=)g Fy(\024)2462 1457
y Fv(1)2501 1442 y Fw(j)p Fy(q)2572 1457 y Fx(j)2609
1442 y Fw(j)2637 1401 y Fv(2)p Fx(d)2637 1470 y(L)2685
1451 y Fq(2)p Fn(d)2772 1442 y Fy(:)100 1664 y FA(W)-8
b(e)33 b(also)f(deduce,)i(b)m(y)g(\(2.25\),)e(H\177)-49
b(older)33 b(inequalit)m(y)h(and)f(orthogonalit)m(y)640
1883 y Fw(K)716 1898 y Fu(1)813 1883 y FA(+)22 b(1)g(+)g
Fw(j)p Fy(q)1151 1898 y Fx(j)1187 1883 y Fw(j)1215 1898
y Fx(H)1273 1907 y Fq(1)1394 1883 y Fw(\025)84 b Fy(\024)1611
1898 y Fv(2)1651 1772 y Fz(\020)1727 1747 y(Z)1782 1973
y Fp(T)1826 1954 y Fq(2)1882 1772 y Fz(\020)1941 1883
y Fy(q)1984 1898 y Fv(+)p Fx(;j)2118 1883 y FA(+)22 b
Fy(q)2259 1898 y Fv(0)p Fx(;j)2372 1883 y FA(+)g Fy(q)2513
1898 y Fu(\000)p Fx(;j)2625 1772 y Fz(\021)2684 1795
y Fv(2)2724 1772 y Fz(\021)2783 1795 y Fx(d)1395 2146
y FA(=)84 b Fy(\024)1611 2161 y Fv(2)1651 2035 y Fz(\020)1727
2010 y(Z)1782 2236 y Fp(T)1826 2217 y Fq(2)1882 2146
y Fy(q)1929 2105 y Fv(2)1925 2170 y(+)p Fx(;j)2058 2146
y FA(+)22 b Fy(q)2203 2105 y Fv(2)2199 2170 y(0)p Fx(;j)2313
2146 y FA(+)g Fy(q)2458 2105 y Fv(2)2454 2170 y Fu(\000)p
Fx(;j)2565 2035 y Fz(\021)2625 2057 y Fx(d)2693 2146
y Fw(\025)28 b Fy(\024)2854 2161 y Fv(3)2926 2146 y FA(\()p
Fy(q)3007 2161 y Fv(0)p Fx(;j)3099 2146 y FA(\))3137
2105 y Fv(2)p Fx(d)100 2363 y FA(and)k(therefore)100
2559 y(\(2.26\))1004 b Fw(j)p Fy(q)1425 2574 y Fv(0)p
Fx(;j)1516 2559 y Fw(j)28 b(\024)g Fy(\024)1733 2574
y Fv(4)1772 2448 y Fz(\020)1832 2559 y FA(1)22 b(+)g
Fw(j)p Fy(q)2072 2574 y Fx(j)2108 2559 y Fw(j)2136 2574
y Fx(H)2194 2583 y Fq(1)2233 2448 y Fz(\021)2292 2470
y Fv(1)p Fx(=)p Fv(2)p Fx(d)2471 2559 y Fy(:)199 2756
y FA(By)33 b(\(2.18\)-\(2.19\))e(and)i(H\177)-49 b(older)33
b(inequalit)m(y)107 2949 y Fz(e)100 2974 y FA(\010)170
2933 y Fu(0)170 2999 y Fx(";N)290 2974 y FA(\()p Fy(q)371
2989 y Fx(j)407 2974 y FA(\)[)p Fy(q)515 2989 y Fv(+)p
Fx(;j)627 2974 y FA(])83 b(=)h Fy(\013)959 2989 y Fv(+)1018
2974 y Fw(j)p Fy(q)1089 2989 y Fv(+)p Fx(;j)1200 2974
y Fw(j)1228 2933 y Fv(2)1228 2999 y Fx(H)1286 3008 y
Fq(1)1347 2974 y Fw(\000)1446 2839 y Fz(Z)1502 3064 y
Fp(T)1546 3045 y Fq(2)1601 2974 y Fy(a)1652 2989 y Fv(2)p
Fx(d)p Fu(\000)p Fv(1)1818 2974 y FA(\()p Fy(')1920 2989
y Fv(1)1960 2974 y FA(\))p Fy(q)2045 2933 y Fv(2)p Fx(d)p
Fu(\000)p Fv(1)2041 3000 y Fx(j)2210 2974 y Fy(q)2253
2989 y Fv(+)p Fx(;j)2387 2974 y FA(+)2508 2949 y Fz(e)2485
2974 y Fw(R)2569 2933 y Fu(0)2569 2999 y Fx(";N)2689
2974 y FA(\()p Fy(q)2770 2989 y Fx(j)2806 2974 y FA(\)[)p
Fy(q)2914 2989 y Fv(+)p Fx(;j)3026 2974 y FA(])737 3237
y Fw(\025)f Fy(\013)959 3252 y Fv(+)1018 3237 y Fw(j)p
Fy(q)1089 3252 y Fv(+)p Fx(;j)1200 3237 y Fw(j)1228 3196
y Fv(2)1228 3262 y Fx(H)1286 3271 y Fq(1)1347 3237 y
Fw(\000)22 b Fy(\024)1502 3252 y Fv(5)1542 3237 y Fw(j)p
Fy(q)1613 3252 y Fv(+)p Fx(;j)1724 3237 y Fw(j)1752 3252
y Fx(H)1810 3261 y Fq(1)1865 3101 y Fz(Z)1920 3327 y
Fp(T)1964 3308 y Fq(2)2020 3237 y Fw(j)p Fy(q)2091 3252
y Fx(j)2127 3237 y Fw(j)2155 3196 y Fv(2)p Fx(d)p Fu(\000)p
Fv(1)2343 3237 y Fw(\000)h FA(\()p Fy(\016)j FA(+)c Fy(\015)2704
3196 y Fu(\000)p Fv(1)2798 3237 y Fw(j)p Fy(")p Fw(j)f
FA(+)h Fy(N)3107 3196 y Fu(\000)p Fv(2)3202 3237 y FA(\))p
Fy(C)3310 3252 y Fv(2)3349 3237 y FA(\()p Fy(R)q FA(\))p
Fw(j)p Fy(q)3571 3252 y Fv(+)p Fx(;j)3682 3237 y Fw(j)3710
3252 y Fx(H)3768 3261 y Fq(1)737 3479 y Fw(\025)83 b
Fy(\024)953 3494 y Fv(6)993 3479 y Fw(j)p Fy(q)1064 3494
y Fv(+)p Fx(;j)1175 3479 y Fw(j)1203 3494 y Fx(H)1261
3503 y Fq(1)1299 3368 y Fz(\020)1359 3479 y Fw(j)p Fy(q)1430
3494 y Fv(+)p Fx(;j)1540 3479 y Fw(j)1568 3494 y Fx(H)1626
3503 y Fq(1)1687 3479 y Fw(\000)23 b(j)p Fy(q)1858 3494
y Fx(j)1894 3479 y Fw(j)1922 3438 y Fv(2)p Fx(d)p Fu(\000)p
Fv(1)1922 3514 y Fx(L)1970 3495 y Fq(2)p Fn(d)2110 3479
y Fw(\000)f FA(1)2258 3368 y Fz(\021)2350 3479 y Fy(:)-2224
b FA(\(2.27\))100 3699 y(By)33 b(\(2.27\))f(and)g(\(2.25\),)g(using)h
(that)1507 3673 y Fz(e)1500 3699 y FA(\010)1570 3662
y Fu(0)1570 3724 y Fx(";N)1690 3699 y FA(\()p Fy(q)1771
3714 y Fx(j)1807 3699 y FA(\))28 b Fw(!)f FA(0)33 b(and)f(simple)i
(inequalities,)h(w)m(e)f(conclude)1254 3909 y Fw(j)p
Fy(q)1325 3924 y Fv(+)p Fx(;j)1435 3909 y Fw(j)1463 3924
y Fx(H)1521 3933 y Fq(1)1588 3909 y Fw(\024)28 b Fy(\024)1749
3924 y Fv(7)1788 3798 y Fz(\020)1848 3909 y FA(1)22 b(+)g
Fw(j)p Fy(q)2088 3924 y Fx(j)2124 3909 y Fw(j)2152 3858
y Fv(\(2)p Fx(d)p Fu(\000)p Fv(1\))p Fx(=)p Fv(2)p Fx(d)2152
3936 y(H)2210 3945 y Fq(1)2480 3798 y Fz(\021)2572 3909
y Fy(:)100 4129 y FA(Estimating)33 b(analogously)1137
4104 y Fz(e)1129 4129 y FA(\010)1199 4093 y Fu(0)1199
4154 y Fx(";N)1319 4129 y FA(\()p Fy(q)1400 4144 y Fx(j)1437
4129 y FA(\)[)p Fy(q)1545 4144 y Fu(\000)p Fx(;j)1656
4129 y FA(])g(w)m(e)g(deriv)m(e)1283 4339 y Fw(j)p Fy(q)1354
4354 y Fu(\000)p Fx(;j)1465 4339 y Fw(j)1493 4354 y Fx(H)1551
4363 y Fq(1)1617 4339 y Fw(\024)c Fy(\024)1779 4354 y
Fv(8)1818 4229 y Fz(\020)1878 4339 y FA(1)22 b(+)g Fw(j)p
Fy(q)2118 4354 y Fx(j)2154 4339 y Fw(j)2182 4288 y Fv(\(2)p
Fx(d)p Fu(\000)p Fv(1\))p Fx(=)p Fv(2)p Fx(d)2182 4366
y(H)2240 4375 y Fq(1)2509 4229 y Fz(\021)100 4536 y FA(and)32
b(b)m(y)i(\(2.26\))e(w)m(e)h(\014nally)h(deduce)503 4734
y Fw(j)p Fy(q)574 4749 y Fx(j)610 4734 y Fw(j)638 4749
y Fx(H)696 4758 y Fq(1)762 4734 y FA(=)28 b Fw(j)p Fy(q)937
4749 y Fv(0)p Fx(;j)1028 4734 y Fw(j)22 b FA(+)g Fw(j)p
Fy(q)1247 4749 y Fv(+)p Fx(;j)1358 4734 y Fw(j)1386 4749
y Fx(H)1444 4758 y Fq(1)1505 4734 y FA(+)g Fw(j)p Fy(q)1674
4749 y Fu(\000)p Fx(;j)1785 4734 y Fw(j)1813 4749 y Fx(H)1871
4758 y Fq(1)1937 4734 y Fw(\024)28 b Fy(\024)2098 4749
y Fv(9)2137 4623 y Fz(\020)2197 4734 y FA(1)22 b(+)g
Fw(j)p Fy(q)2437 4749 y Fx(j)2473 4734 y Fw(j)2501 4683
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4734 y Fw(j)2903 4683 y Fv(\(2)p Fx(d)p Fu(\000)p Fv(1\))p
Fx(=)p Fv(2)p Fx(d)2903 4761 y(H)2961 4770 y Fq(1)3230
4623 y Fz(\021)3322 4734 y Fy(:)100 4931 y FA(W)-8 b(e)31
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y Fw(j)1011 4946 y Fx(H)1069 4955 y Fq(1)1135 4931 y
Fw(\024)e Fy(R)1315 4946 y Fu(\003)1385 4931 y FA(for)i(a)g(suitable)h
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4931 y FA(indep)s(enden)m(t)h(of)e Fy(")p FA(,)g Fy(N)10
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100 5242 47 4 v 100 5297 a Fy(q)146 5320 y Fv(1)219 5297
y FA(of)338 5272 y Fz(e)330 5297 y FA(\010)400 5312 y
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5320 y Fv(1)890 5297 y Fw(j)918 5312 y Fx(H)976 5321
y Fq(1)1043 5297 y Fw(\024)29 b Fy(R)1223 5312 y Fu(\003)1263
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100 5358 V 100 5413 a Fy(q)146 5437 y Fv(1)214 5413 y
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%%Page: 15 15
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281 y Fv(1)2517 266 y FA(=)28 b(1.)100 456 y Fo(Pr)-5
b(o)g(of)34 b(of)h(The)-5 b(or)g(em)34 b(A)h(for)f Fy(!)1204
471 y Fv(1)1271 456 y FA(=)28 b(1)p Fo(.)41 b FA(Let)33
b(us)g(\014x)876 608 y(\026)855 634 y Fy(R)c FA(:=)e
Fy(R)1162 649 y Fu(\003)1224 634 y FA(+)22 b(1)195 b(and)33
b(tak)m(e)196 b Fw(j)p Fy(")p Fw(j)p Fy(\015)2288 592
y Fu(\000)p Fv(1)2409 634 y Fw(\024)28 b Fy(")2560 649
y Fv(2)2599 634 y FA(\()2658 608 y(\026)2637 634 y Fy(R)q
FA(\))f(:=)p 2908 579 46 4 v 28 w Fy(")16 b(:)100 811
y FA(Set)296 786 y(\026)268 811 y Fy(N)38 b FA(:=)27
b Fy(N)592 826 y Fv(2)632 811 y FA(\()691 786 y(\026)670
811 y Fy(R)q FA(\))h Fw(\025)g Fy(N)994 826 y Fv(0)1033
811 y FA(\()1093 786 y(\026)1071 811 y Fy(R)q FA(\).)199
927 y(Applying)34 b(Lemma)f(2.3)f(w)m(e)i(obtain,)e(for)1596
1147 y(0)27 b Fy(<)h(\033)k Fw(\024)2088 1080 y FA(1)p
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1218 y FA(\()2155 1192 y(\026)2133 1218 y Fy(R)q FA(\))100
1393 y(a)c(solution)g(\()p Fy(q)622 1408 y Fv(2)662 1393
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b Fw(2)i FA(\()p Fy(Q)1347 1408 y Fv(2)1387 1393 y FA(\()1453
1368 y(\026)1425 1393 y Fy(N)10 b FA(\))j Fw(\010)g Fy(P)h
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y FA(of)28 b(the)h(\()p Fy(Q)2456 1408 y Fv(2)2495 1393
y FA(\)-\()p Fy(P)14 b FA(\))27 b(equations)j Fw(8j)p
Fy(q)3305 1408 y Fv(1)3344 1393 y Fw(j)3372 1408 y Fx(H)3430
1417 y Fq(1)3496 1393 y Fw(\024)f FA(2)3672 1368 y(\026)3651
1393 y Fy(R)p FA(.)100 1527 y(By)k(Lemma)h(2.6)f(the)h(extended)h
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b(\026)-56 b Fy(q)3308 1542 y Fv(1)3376 1527 y Fw(6)p
FA(=)29 b(0)k(with)100 1660 y Fw(j)7 b FA(\026)-56 b
Fy(q)171 1675 y Fv(1)210 1660 y Fw(j)238 1675 y Fx(H)296
1684 y Fq(1)374 1660 y Fw(\024)40 b Fy(R)565 1675 y Fu(\003)645
1660 y Fy(<)782 1635 y FA(\026)761 1660 y Fy(R)q FA(.)64
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1680 y Fx(R)3300 1660 y FA(w)m(e)h(get,)h(b)m(y)100 1776
y(Lemma)c(2.4,)i(the)f(existence)h(of)e(a)g(non)m(trivial)h(w)m(eak)h
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2852 1721 V Fy(q)2899 1800 y Fv(1)2938 1776 y FA(\))g(+)g
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1776 y FA(\))38 b Fw(2)g(H)3541 1791 y Fx(\033)n(;s)3674
1776 y FA(of)100 1892 y(equation)33 b(\(2.1\).)43 b(Finally)681
2101 y Fy(u)27 b FA(=)h Fw(j)p Fy(")p Fw(j)970 2060 y
Fv(1)p Fx(=)p Fv(2\()p Fx(d)p Fu(\000)p Fv(1\))1260 1990
y Fz(h)p 1307 2046 V 111 x Fy(q)1354 2125 y Fv(1)1416
2101 y FA(+)22 b Fy(q)1557 2116 y Fv(2)1596 2101 y FA(\()p
1634 2046 V Fy(q)1681 2125 y Fv(1)1721 2101 y FA(\))g(+)g
Fy(p)p FA(\()p 1966 2046 V Fy(q)2012 2125 y Fv(1)2052
2101 y FA(\))2090 1990 y Fz(i)2165 2101 y Fw(\021)28
b(j)p Fy(")p Fw(j)2372 2060 y Fv(1)p Fx(=)p Fv(2\()p
Fx(d)p Fu(\000)p Fv(1\))2662 1990 y Fz(h)p 2709 2046
V 111 x Fy(q)2756 2125 y Fx(")2815 2101 y FA(+)22 b Fy(p)p
FA(\()p 3000 2046 V Fy(q)3047 2125 y Fv(1)3086 2101 y
FA(\))3124 1990 y Fz(i)100 2310 y FA(solv)m(es)34 b(equation)f
(\(1.4\).)199 2426 y(The)44 b(solution)p 789 2371 V 43
w Fy(q)836 2449 y Fx(")918 2426 y FA(:=)p 1066 2371 V
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2449 y Fv(1)1493 2426 y FA(\))43 b(of)f(the)h(\()p Fy(Q)p
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2505 y Fu(\000)p Fv(1)1266 2571 y(1)1361 2546 y FA(,)h(see)g(in)g
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b(\026)-57 b Fy(p)28 b FA(:=)f Fy(p)p FA(\()p 748 2607
V Fy(q)795 2686 y Fv(1)834 2662 y FA(\))33 b(solv)m(es)100
2871 y(\(2.28\))399 2760 y Fz(\020)458 2871 y Fy(@)514
2830 y Fv(2)509 2896 y Fx(')555 2905 y Fq(1)596 2871
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2871 y FA(\(2)q(+)q Fy(")p FA(\))p Fy(@)1957 2830 y Fv(2)1952
2896 y Fx(')1998 2905 y Fq(2)2045 2871 y FA(\026)-58
b Fy(p)q FA(+)q(\005)2236 2886 y Fx(P)2295 2871 y Fy(f)11
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y Fx(\033)3018 2867 y Fm(0)3041 2886 y Fx(;s)3195 2871
y Fw(8)p FA(0)g Fy(<)f(\033)3489 2830 y Fu(0)3540 2871
y Fy(<)h(\033)100 3084 y FA(and)k(the)h(eigen)m(v)-5
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3099 y Fx(')1902 3108 y Fq(2)1974 3084 y FA(restricted)34
b(to)e Fy(P)46 b FA(satisfy)-8 b(,)33 b(for)f Fy(")c
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y Fz(\014)758 3278 y(\014)758 3338 y(\014)791 3333 y
Fy(l)820 3348 y Fv(1)859 3222 y Fz(h)906 3333 y FA(\()p
Fy(l)973 3348 y Fv(1)1035 3333 y FA(+)22 b(2)p Fy(l)1211
3348 y Fv(2)1250 3333 y FA(\))g(+)g Fy(")p FA(2)p Fy(l)1532
3348 y Fv(2)1571 3222 y Fz(i)1618 3218 y(\014)1618 3278
y(\014)1618 3338 y(\014)1679 3333 y Fw(\025)28 b Fy(\015)1875
3265 y Fw(j)p Fy(l)1932 3280 y Fv(1)1971 3265 y Fw(j)p
1850 3310 173 4 v 1850 3401 a FA(2)p Fw(j)p Fy(l)1956
3416 y Fv(2)1995 3401 y Fw(j)2228 3333 y(8)17 b Fy(l)2329
3348 y Fv(1)2391 3333 y FA(+)22 b(2)p Fy(l)2567 3348
y Fv(2)2633 3333 y Fw(6)p FA(=)28 b(0)p Fy(;)33 b(l)2875
3348 y Fv(2)2942 3333 y Fw(6)p FA(=)28 b(0)100 3574 y(and)35
b(w)m(e)h(deduce)g(that)43 b(\026)-57 b Fy(p)32 b Fw(2)g(H)1242
3589 y Fx(\033)1284 3570 y Fm(00)1325 3589 y Fx(;s)p
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3538 y Fu(00)2117 3574 y Fy(<)h(\033)2284 3538 y Fu(0)2342
3574 y FA(and)j Fw(j)p Fy(@)2613 3589 y Fx(')2659 3598
y Fq(1)2706 3574 y FA(\026)-57 b Fy(p)p Fw(j)2775 3589
y Fx(\033)2817 3570 y Fm(00)2858 3589 y Fx(;s)2946 3574
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3858 y Fv(2)255 3924 y Fx(')301 3933 y Fq(1)348 3899
y FA(\026)-57 b Fy(p)28 b FA(=)f Fw(\000)p FA(2\(1)22
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Fq(2)1073 3899 y Fy(@)1124 3914 y Fx(')1170 3923 y Fq(1)1218
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b(\033)3562 3858 y Fu(00)3621 3899 y Fy(:)100 4112 y
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Fw(j)p Fy(")p Fw(j)p Fy(\015)1874 4076 y Fu(\000)p Fv(1)1966
4112 y FA(\).)44 b(\(1.5\))32 b(follo)m(ws)h(with)k(\026)-53
b Fy(s)28 b FA(:=)g Fy(s)22 b FA(+)g(2)27 b Fy(>)h FA(2.)199
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h(Lemma)g(2.5)1695 4978 y(~)1684 5003 y(\010)1754 5018
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b Fy(q)840 5135 y Fv(1)919 5120 y Fy(=)-61 b Fw(2)29
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%%Page: 16 16
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3930 y Fy(n)p 1461 3974 86 4 v 1461 4065 a(m)1556 3997
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602 4387 1423 4 v 1251 4478 a Fy(m)1336 4450 y Fv(2)2062
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%%Page: 17 17
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b Ft(Q)199 1925 y FA(W)-8 b(e)44 b(no)m(w)g(lo)s(ok)f(for)g(solutions)h
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2391 y Fy(;)17 b Fw(H)3358 2406 y Fx(\033)n(;)p 3416
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%%Page: 20 20
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(\(4.6\).)p eop end
%%Page: 23 23
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964 y FA(\()p Fy(q)3362 979 y Fv(1)3402 964 y FA(\)[)p
Fy(h)p FA(])3550 853 y Fz(\021)100 1212 y FA(and)32 b(using)i(\(2.7\),)
e(\(2.6\))g(and)g(the)h(Banac)m(h)h(algebra)e(prop)s(ert)m(y)h(of)f
Fw(H)2678 1227 y Fx(\033)n(;s)603 1464 y Fw(j)p Fy(q)678
1423 y Fu(0)674 1489 y Fv(2)713 1464 y FA(\()p Fy(q)794
1479 y Fv(1)834 1464 y FA(\)[)p Fy(h)p FA(])p Fw(j)1010
1479 y Fx(\033)n(;s)1132 1464 y Fw(\024)d Fy(C)7 b FA(\()p
Fy(R)q FA(\))p Fy(N)1554 1423 y Fu(\000)p Fv(2)1648 1353
y Fz(\020)1708 1464 y Fw(j)p Fy(h)p Fw(j)1820 1479 y
Fx(\033)n(;s)1937 1464 y FA(+)22 b Fw(j)p Fy(q)2110 1423
y Fu(0)2106 1489 y Fv(2)2145 1464 y FA(\()p Fy(q)2226
1479 y Fv(1)2266 1464 y FA(\)[)p Fy(h)p FA(])p Fw(j)2442
1479 y Fx(\033)n(;s)2559 1464 y FA(+)g Fw(j)p Fy(p)2734
1423 y Fu(0)2757 1464 y FA(\()p Fy(q)2838 1479 y Fv(1)2877
1464 y FA(\)[)p Fy(h)p FA(])p Fw(j)3053 1479 y Fx(\033)n(;s)3149
1353 y Fz(\021)603 1681 y Fw(j)p Fy(p)680 1639 y Fu(0)703
1681 y FA(\()p Fy(q)784 1696 y Fv(1)823 1681 y FA(\)[)p
Fy(h)p FA(])p Fw(j)999 1696 y Fx(\033)n(;s)1122 1681
y Fw(\024)28 b Fy(C)7 b FA(\()p Fy(R)q FA(\))p Fw(j)p
Fy(")p Fw(j)p Fy(\015)1613 1639 y Fu(\000)p Fv(1)1706
1570 y Fz(\020)1766 1681 y Fw(j)p Fy(h)p Fw(j)1878 1696
y Fx(\033)n(;s)1995 1681 y FA(+)22 b Fw(j)p Fy(q)2168
1639 y Fu(0)2164 1705 y Fv(2)2203 1681 y FA(\()p Fy(q)2284
1696 y Fv(1)2324 1681 y FA(\)[)p Fy(h)p FA(])p Fw(j)2500
1696 y Fx(\033)n(;s)2617 1681 y FA(+)g Fw(j)p Fy(p)2792
1639 y Fu(0)2815 1681 y FA(\()p Fy(q)2896 1696 y Fv(1)2936
1681 y FA(\)[)p Fy(h)p FA(])p Fw(j)3112 1696 y Fx(\033)n(;s)3207
1570 y Fz(\021)100 1929 y FA(whic)m(h)34 b(implies)g(the)f(b)s(ounds)g
(\(2.11\))f(since)976 2268 y(det)1128 2063 y Fz(\014)1128
2123 y(\014)1128 2183 y(\014)1128 2243 y(\014)1128 2303
y(\014)1128 2362 y(\014)1203 2150 y FA(1)22 b Fw(\000)g
Fy(C)7 b FA(\()p Fy(R)q FA(\))p Fy(N)1689 2114 y Fu(\000)p
Fv(2)1950 2150 y Fw(\000)p Fy(C)g FA(\()p Fy(R)q FA(\))p
Fy(N)2343 2114 y Fu(\000)p Fv(2)1203 2382 y Fw(\000)p
Fy(C)g FA(\()p Fy(R)q FA(\))p Fw(j)p Fy(")p Fw(j)p Fy(\015)1666
2346 y Fu(\000)p Fv(1)1950 2382 y FA(1)22 b Fw(\000)h
Fy(C)7 b FA(\()p Fy(R)q FA(\))p Fw(j)p Fy(")p Fw(j)p
Fy(\015)2507 2346 y Fu(\000)p Fv(1)2641 2063 y Fz(\014)2641
2123 y(\014)2641 2183 y(\014)2641 2243 y(\014)2641 2303
y(\014)2641 2362 y(\014)2702 2268 y Fw(\025)2817 2200
y FA(1)p 2817 2245 49 4 v 2817 2336 a(2)100 2610 y(for)32
b Fy(C)7 b FA(\()p Fy(R)q FA(\)\()p Fw(j)p Fy(")p Fw(j)p
Fy(\015)673 2574 y Fu(\000)p Fv(1)788 2610 y FA(+)22
b Fy(N)974 2574 y Fu(\000)p Fv(2)1069 2610 y FA(\))32
b(small)i(enough)f(and)f(\(4.3\).)100 2928 y Fc(Pro)s(of)g(of)g(Lemma)h
(2.4.)86 b FA(By)33 b(\(2.4\),)f(\(2.5\))h(w)m(e)g(ha)m(v)m(e)h(that,)e
(at)h Fy(u)27 b FA(:=)h Fy(q)2694 2943 y Fv(1)2755 2928
y FA(+)22 b Fy(q)2896 2943 y Fv(2)2936 2928 y FA(\()p
Fy(q)3017 2943 y Fv(1)3057 2928 y FA(\))g(+)g Fy(p)p
FA(\()p Fy(q)3345 2943 y Fv(1)3384 2928 y FA(\),)100
3146 y(\(4.7\))366 b Fy(d)p FA(\011)794 3161 y Fx(")830
3146 y FA(\()p Fy(u)p FA(\)[)p Fy(h)p FA(])27 b(=)h(0)49
b Fw(8)p Fy(h)28 b Fw(2)g Fy(Q)1611 3161 y Fv(2)1846
3146 y FA(and)195 b Fy(d)p FA(\011)2325 3161 y Fx(")2361
3146 y FA(\()p Fy(u)p FA(\)[)p Fy(h)p FA(])28 b(=)f(0)49
b Fw(8)p Fy(h)28 b Fw(2)g Fy(P)j(:)100 3363 y FA(Since)i
Fy(q)401 3327 y Fu(0)397 3387 y Fv(2)437 3363 y FA(\()p
Fy(q)518 3378 y Fv(1)558 3363 y FA(\)[)p Fy(k)s FA(])28
b Fw(2)g Fy(Q)903 3378 y Fv(2)975 3363 y FA(and)33 b
Fy(p)1214 3327 y Fu(0)1237 3363 y FA(\()p Fy(q)1318 3378
y Fv(1)1357 3363 y FA(\)[)p Fy(k)s FA(])28 b Fw(2)g Fy(P)46
b Fw(8)p Fy(k)31 b Fw(2)e Fy(Q)2043 3378 y Fv(1)2082
3363 y FA(,)k(w)m(e)g(deduce)422 3611 y Fy(d)p FA(\010)543
3626 y Fx(";N)662 3611 y FA(\()p Fy(q)743 3626 y Fv(1)783
3611 y FA(\)[)p Fy(k)s FA(])28 b(=)f Fy(d)p FA(\011)1187
3626 y Fx(")1224 3611 y FA(\()p Fy(u)p FA(\))1356 3501
y Fz(h)1402 3611 y Fy(h)22 b FA(+)g Fy(q)1625 3570 y
Fu(0)1621 3636 y Fv(2)1661 3611 y FA(\()p Fy(q)1742 3626
y Fv(1)1782 3611 y FA(\)[)p Fy(k)s FA(])g(+)g Fy(p)2097
3570 y Fu(0)2120 3611 y FA(\()p Fy(q)2201 3626 y Fv(1)2241
3611 y FA(\)[)p Fy(k)s FA(])2387 3501 y Fz(i)2462 3611
y FA(=)27 b Fy(d)p FA(\011)2692 3626 y Fx(")2729 3611
y FA(\()p Fy(u)p FA(\)[)p Fy(k)s FA(])114 b Fw(8)p Fy(k)31
b Fw(2)d Fy(Q)3391 3626 y Fv(1)100 3860 y FA(and)k(therefore)h
Fy(u)28 b FA(:=)f Fy(q)955 3875 y Fv(1)1017 3860 y FA(+)22
b Fy(p)p FA(\()p Fy(q)1245 3875 y Fv(1)1285 3860 y FA(\))g(+)g
Fy(q)1486 3875 y Fv(2)1526 3860 y FA(\()p Fy(q)1607 3875
y Fv(1)1646 3860 y FA(\))33 b(solv)m(es)h(also)f(the)g(\()p
Fy(Q)2473 3875 y Fv(1)2512 3860 y FA(\)-equation)g(\(2.3\).)199
3994 y(W)-8 b(rite)33 b(\011)546 4009 y Fx(")583 3994
y FA(\()p Fy(u)p FA(\))27 b(=)g(\011)921 3943 y Fv(\(2\))921
4004 y Fx(")1016 3994 y FA(\()p Fy(u)p FA(\))21 b Fw(\000)i
Fy(")1332 3914 y Fz(R)1378 4029 y Fp(T)1422 4010 y Fq(2)1478
3994 y Fy(F)14 b FA(\()p Fy(')1657 4009 y Fv(1)1696 3994
y Fy(;)j(u;)g(\016)t FA(\))31 b(where)566 4280 y(\011)642
4239 y Fv(\(2\))642 4305 y Fx(")736 4280 y FA(\()p Fy(u)p
FA(\))d(:=)1026 4144 y Fz(Z)1082 4370 y Fp(T)1126 4351
y Fq(2)1191 4213 y FA(1)p 1191 4257 V 1191 4348 a(2)1250
4280 y(\()p Fy(@)1339 4295 y Fx(')1385 4304 y Fq(1)1424
4280 y Fy(u)p FA(\))1518 4239 y Fv(2)1579 4280 y FA(+)22
b(\(1)g(+)g Fy(")p FA(\)\()p Fy(@)2057 4295 y Fx(')2103
4304 y Fq(1)2142 4280 y Fy(u)p FA(\)\()p Fy(@)2325 4295
y Fx(')2371 4304 y Fq(2)2409 4280 y Fy(u)p FA(\))g(+)2633
4213 y Fy(")p FA(\(2)f(+)h Fy(")p FA(\))p 2633 4257 337
4 v 2777 4348 a(2)2979 4280 y(\()p Fy(@)3068 4295 y Fx(')3114
4304 y Fq(2)3153 4280 y Fy(u)p FA(\))3247 4239 y Fv(2)100
4548 y FA(is)33 b(an)f(homogeneous)i(functional)f(of)f(degree)h(t)m(w)m
(o.)45 b(By)33 b(homogeneit)m(y:)100 4813 y(\(4.8\))758
b(\011)1135 4828 y Fx(")1172 4813 y FA(\()p Fy(u)p FA(\))27
b(=)1444 4745 y(1)p 1444 4790 49 4 v 1444 4881 a(2)1503
4813 y Fy(d)p FA(\011)1630 4772 y Fv(\(2\))1630 4837
y Fx(")1724 4813 y FA(\()p Fy(u)p FA(\)[)p Fy(u)p FA(])21
b Fw(\000)i Fy(")2150 4677 y Fz(Z)2204 4903 y Fp(T)2248
4884 y Fq(2)2304 4813 y Fy(F)14 b FA(\()p Fy(')2483 4828
y Fv(1)2522 4813 y Fy(;)j(u;)g(\016)t FA(\))g Fy(:)100
5087 y FA(By)33 b(\(2.4\),)f(\(2.5\))g(\(i.e.)44 b(\(4.7\)\))100
5356 y(\(4.9\))147 b Fy(d)p FA(\011)575 5315 y Fv(\(2\))575
5381 y Fx(")669 5356 y FA(\()p Fy(q)750 5371 y Fv(1)812
5356 y FA(+)22 b Fy(q)953 5371 y Fv(2)993 5356 y FA(\()p
Fy(q)1074 5371 y Fv(1)1113 5356 y FA(\))g(+)g Fy(p)p
FA(\()p Fy(q)1401 5371 y Fv(1)1441 5356 y FA(\)\)[)p
Fy(q)1587 5371 y Fv(2)1626 5356 y FA(\()p Fy(q)1707 5371
y Fv(1)1747 5356 y FA(\))g(+)g Fy(p)p FA(\()p Fy(q)2035
5371 y Fv(1)2075 5356 y FA(\)])27 b(=)h Fy(")2334 5221
y Fz(Z)2389 5446 y Fp(T)2433 5427 y Fq(2)2488 5356 y
Fy(f)11 b FA(\()p Fy(')2649 5371 y Fv(1)2688 5356 y Fy(;)17
b(u;)g(\016)t FA(\)\()p Fy(q)2998 5371 y Fv(2)3037 5356
y FA(\()p Fy(q)3118 5371 y Fv(1)3157 5356 y FA(\))22
b(+)g Fy(p)p FA(\()p Fy(q)3445 5371 y Fv(1)3485 5356
y FA(\)\))17 b Fy(:)p eop end
%%Page: 24 24
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b(in)g(\(4.8\))f(w)m(e)i(obtain,)e(at)h Fy(u)27 b FA(=)g
Fy(q)1828 281 y Fv(1)1890 266 y FA(+)22 b Fy(q)2031 281
y Fv(2)2071 266 y FA(\()p Fy(q)2152 281 y Fv(1)2192 266
y FA(\))g(+)g Fy(p)p FA(\()p Fy(q)2480 281 y Fv(1)2519
266 y FA(\))100 480 y(\010)170 495 y Fx(";N)290 480 y
FA(\()p Fy(q)371 495 y Fv(1)410 480 y FA(\))83 b(=)g(\011)766
495 y Fx(")803 480 y FA(\()p Fy(q)884 495 y Fv(1)945
480 y FA(+)22 b Fy(p)p FA(\()p Fy(q)1173 495 y Fv(1)1213
480 y FA(\))g(+)g Fy(q)1414 495 y Fv(2)1454 480 y FA(\()p
Fy(q)1535 495 y Fv(1)1574 480 y FA(\)\))28 b(=)1791 413
y(1)p 1791 457 49 4 v 1791 548 a(2)1850 480 y Fy(d)p
FA(\011)1977 439 y Fv(\(2\))1977 505 y Fx(")2071 480
y FA(\()p Fy(u)p FA(\)[)p Fy(q)2273 495 y Fv(1)2334 480
y FA(+)22 b Fy(p)p FA(\()p Fy(q)2562 495 y Fv(1)2602
480 y FA(\))g(+)g Fy(q)2803 495 y Fv(2)2843 480 y FA(\()p
Fy(q)2924 495 y Fv(1)2963 480 y FA(\)])h Fw(\000)f Fy(")3213
345 y Fz(Z)3268 570 y Fp(T)3312 551 y Fq(2)3367 480 y
Fy(F)14 b FA(\()p Fy(')3546 495 y Fv(1)3585 480 y Fy(;)j(u;)g(\016)t
FA(\))531 743 y(=)700 676 y(1)p 700 720 V 700 811 a(2)759
743 y Fy(d)p FA(\011)886 702 y Fv(\(2\))886 768 y Fx(")980
743 y FA(\()p Fy(q)1061 758 y Fv(1)1100 743 y FA(\)[)p
Fy(q)1208 758 y Fv(1)1248 743 y FA(])22 b Fw(\000)h Fy(")1460
607 y Fz(Z)1514 833 y Fp(T)1558 814 y Fq(2)1614 743 y
Fy(F)14 b FA(\()p Fy(')1793 758 y Fv(1)1832 743 y Fy(;)j(u;)g(\016)t
FA(\))k(+)2190 676 y(1)p 2190 720 V 2190 811 a(2)2249
743 y Fy(f)11 b FA(\()p Fy(')2410 758 y Fv(1)2449 743
y Fy(;)17 b(u;)g(\016)t FA(\)\()p Fy(q)2759 758 y Fv(2)2797
743 y FA(\()p Fy(q)2878 758 y Fv(1)2918 743 y FA(\))22
b(+)g Fy(p)p FA(\()p Fy(q)3206 758 y Fv(1)3245 743 y
FA(\)\))531 1012 y(=)83 b(\011)766 1027 y Fv(0)805 1012
y FA(\()p Fy(q)886 1027 y Fv(1)926 1012 y FA(\))22 b(+)g
Fy(")1147 877 y Fz(Z)1202 1102 y Fp(T)1246 1083 y Fq(2)1311
945 y FA(\(2)g(+)g Fy(")p FA(\))p 1311 990 291 4 v 1432
1081 a(2)1612 1012 y(\()p Fy(@)1701 1027 y Fx(')1747
1036 y Fq(2)1786 1012 y Fy(q)1829 1027 y Fv(1)1868 1012
y FA(\))1906 971 y Fv(2)1968 1012 y FA(+)g(\()p Fy(@)2155
1027 y Fx(')2201 1036 y Fq(1)2240 1012 y Fy(q)2283 1027
y Fv(1)2323 1012 y FA(\)\()p Fy(@)2450 1027 y Fx(')2496
1036 y Fq(2)2535 1012 y Fy(q)2578 1027 y Fv(1)2618 1012
y FA(\))g Fw(\000)g Fy(F)14 b FA(\()p Fy(')2956 1027
y Fv(1)2995 1012 y Fy(;)j(u;)g(\016)t FA(\))531 1271
y(+)700 1204 y(1)p 700 1248 49 4 v 700 1340 a(2)759 1271
y Fy(f)11 b FA(\()p Fy(')920 1286 y Fv(1)959 1271 y Fy(;)17
b(u;)g(\016)t FA(\)\()p Fy(q)1269 1286 y Fv(2)1307 1271
y FA(\()p Fy(q)1388 1286 y Fv(1)1428 1271 y FA(\))22
b(+)g Fy(p)p FA(\()p Fy(q)1716 1286 y Fv(1)1755 1271
y FA(\)\))28 b(=)f(const)c(+)f Fy(")p FA(\(\000\()p Fy(q)2531
1286 y Fv(1)2571 1271 y FA(\))g(+)g Fw(R)2813 1286 y
Fx(";N)2932 1271 y FA(\()p Fy(q)3013 1286 y Fv(1)3053
1271 y FA(\)\))100 1468 y(b)s(ecause)34 b(\011)537 1483
y Fv(0)576 1468 y FA(\()p Fy(q)657 1483 y Fv(1)696 1468
y FA(\))28 b Fw(\021)33 b FA(const.)199 1585 y(By)g(\(2.10\))f(the)h(b)
s(ounds)g(\(2.14\)-\(2.15\))e(follo)m(w.)100 1792 y Fc(Pro)s(of)i(of)g
(Lemma)h(3.3.)94 b FA(The)34 b(existence)j(of)c Fy(p)p
FA(\()p Fy(\021)t(;)17 b(q)t FA(\))29 b Fw(2)h(H)2282
1807 y Fx(\033)n(;)p 2340 1770 33 3 v(s)2411 1792 y FA(can)k(b)s(e)g
(pro)m(v)m(ed)h(as)f(in)g(Lemma)g(2.3)100 1908 y(using)41
b(the)h(Con)m(traction)f(Mapping)h(Theorem.)70 b(The)42
b(smo)s(othness)h(of)d Fy(p)p FA(\()p Fy(\021)t(;)17
b(q)t FA(\))40 b(follo)m(ws)i(b)m(y)g(the)100 2025 y(Implicit)33
b(F)-8 b(unction)33 b(Theorem)h(since)g Fw(G)6 b FA(\()p
Fy(\021)t(;)17 b(p)p FA(\))32 b(is)h(smo)s(oth)g(in)g
Fy(\021)j FA(and)c Fy(q)t FA(.)199 2141 y(By)27 b(the)g(in)m(v)-5
b(ariance)27 b(of)f(equation)h(\(3.5\))f(under)i(translations)f(in)f
(the)h Fy(')2816 2156 y Fv(2)2882 2141 y FA(v)-5 b(ariable)26
b(the)h(function)100 2257 y Fy(p)p FA(\()p Fy(\021)t(;)17
b(q)t FA(\)\()p Fy(')470 2272 y Fv(1)508 2257 y Fy(;)g(')616
2272 y Fv(2)677 2257 y Fw(\000)23 b Fy(\022)s FA(\))32
b(solv)m(es)410 2451 y Fy(p)p FA(\()p Fy(\021)t(;)17
b(q)t FA(\)\()p Fy(')780 2466 y Fv(1)818 2451 y Fy(;)g(')926
2466 y Fv(2)987 2451 y Fw(\000)23 b Fy(\022)s FA(\))f(+)g
Fy(\021)1345 2409 y Fv(2\()p Fx(d)p Fu(\000)p Fv(1\))1582
2451 y Fw(L)1651 2409 y Fu(\000)p Fv(1)1651 2475 y Fx(")1745
2451 y FA(\005)1818 2466 y Fx(P)1877 2451 y Fy(f)1936
2340 y Fz(\020)1995 2451 y Fy(')2059 2466 y Fv(1)2098
2451 y Fy(;)17 b(q)2185 2466 y Fx(\022)2247 2451 y FA(+)22
b Fy(p)p FA(\()p Fy(\021)t(;)17 b(q)t FA(\)\()p Fy(')2715
2466 y Fv(1)2753 2451 y Fy(;)g(')2861 2466 y Fv(2)2922
2451 y Fw(\000)23 b Fy(\022)s FA(\))p Fy(;)17 b(\021)3204
2340 y Fz(\021)3290 2451 y FA(=)28 b(0)100 2644 y(and,)k(therefore,)i
(b)m(y)f(uniqueness)j(\(3.9\))c(holds.)100 2851 y Fc(Pro)s(of)i(of)g
(Prop)s(osition)h(2.)99 b FA(W)-8 b(rite)35 b Fy(x)p
FA(\()p Fy(E)6 b(;)17 b(t)p FA(\))31 b(=)k Fy(y)t FA(\()p
Fy(!)t FA(\()p Fy(E)6 b FA(\))p Fy(t;)17 b(E)6 b FA(\))33
b(where)j Fy(y)t FA(\()p Fy(';)17 b(E)6 b FA(\))33 b(is)i(2)p
Fy(\031)t FA(-p)s(erio)s(dic)f(in)100 2968 y Fy(')e FA(and)h
Fy(!)t FA(\()p Fy(E)6 b FA(\))27 b(:=)32 b(2)p Fy(\031)t(=T)14
b FA(\()p Fy(E)6 b FA(\).)43 b(The)33 b(functions)h(\()p
Fy(@)1929 2983 y Fx(t)1959 2968 y Fy(x)p FA(\)\()p Fy(E)6
b(;)17 b(t)p FA(\))33 b(and)100 3189 y(\(4.10\))366 b(\()p
Fy(@)805 3204 y Fx(E)865 3189 y Fy(x)p FA(\)\()p Fy(E)6
b(;)17 b(t)p FA(\))28 b(=)g Fy(t)1368 3121 y(d!)t FA(\()p
Fy(E)6 b FA(\))p 1368 3166 269 4 v 1438 3257 a Fy(dE)1647
3189 y FA(\()p Fy(@)1736 3204 y Fx(')1786 3189 y Fy(y)t
FA(\)\()p Fy(!)t FA(\()p Fy(E)g FA(\))p Fy(t;)17 b(E)6
b FA(\))21 b(+)h(\()p Fy(@)2536 3204 y Fx(E)2596 3189
y Fy(y)t FA(\)\()p Fy(!)t FA(\()p Fy(E)6 b FA(\))p Fy(t;)17
b(E)6 b FA(\))100 3386 y(are)33 b(t)m(w)m(o)i(linearly)g(indep)s(enden)
m(t)h(solutions)f(of)e(the)h(linearized)i(equation)e(\(3.13\).)47
b(\()p Fy(@)3297 3401 y Fx(t)3327 3386 y Fy(x)p FA(\)\()p
Fy(E)6 b(;)17 b(t)p FA(\))34 b(is)100 3502 y(2)p Fy(\031)t
FA(-p)s(erio)s(dic)e(while,)i(since)669 3650 y Fy(d!)t
FA(\()p Fy(T)14 b FA(\))p 669 3694 262 4 v 739 3786 a
Fy(dT)968 3717 y FA(=)28 b(2)p Fy(\031)t(T)14 b FA(\()p
Fy(E)6 b FA(\))1405 3676 y Fu(\000)p Fv(2)1509 3650 y
Fy(dE)g FA(\()p Fy(T)14 b FA(\))p 1509 3694 276 4 v 1586
3786 a Fy(dT)1822 3717 y Fw(6)p FA(=)27 b(0)195 b(and)g(\()p
Fy(@)2610 3732 y Fx(')2661 3717 y Fy(y)t FA(\)\()p Fy(';)17
b(E)6 b FA(\))26 b Fw(6\021)i FA(0)100 3914 y(\(if)39
b(not)g Fy(x)p FA(\()p Fy(E)6 b(;)17 b(t)p FA(\))40 b(w)m(ould)h(b)s(e)
e(constan)m(t)i(in)f Fy(t)p FA(\),)h(\()p Fy(@)1925 3929
y Fx(E)1985 3914 y Fy(x)p FA(\)\()p Fy(E)6 b(;)17 b(t)p
FA(\))40 b(is)g(not)f(2)p Fy(\031)t FA(-p)s(erio)s(dic.)64
b(W)-8 b(e)40 b(conclude)100 4031 y(that)35 b(the)i(space)g(of)e
Fy(T)14 b FA(\()p Fy(E)6 b FA(\)-p)s(erio)s(dic)36 b(solutions)h(of)f
(\(3.13\))f(form)h(a)f(1-dimensional)i(linear)f(space)100
4147 y(spanned)e(b)m(y)f(\()p Fy(@)704 4162 y Fx(t)734
4147 y Fy(x)p FA(\)\()p Fy(E)6 b(;)17 b(t)p FA(\).)1636
4359 y Fb(References)100 4516 y FF([AB])41 b(A.)32 b(Am)n(brosetti,)g
(M.)f(Badiale,)g FD(Homo)l(clinics:)47 b(Poinc)l(ar)n(\023)-40
b(e-Melnikov)36 b(typ)l(e)d(r)l(esults)f(via)i(a)g(variational)h(ap-)
270 4616 y(pr)l(o)l(ach)p FF(,)29 b(Annales)f(I.)g(H.)g(P)-7
b(.)27 b(-)g(Analyse)g(nonlin.,)h(v)n(ol.)f(15,)g(n.2,)g(1998,)f(p.)i
(233-252.)100 4716 y([BP])41 b(D.)30 b(Bam)n(busi,)g(S.)g(P)n(aleari,)f
FD(F)-6 b(amilies)33 b(of)g(p)l(erio)l(dic)g(solutions)g(of)f(r)l
(esonant)g(PDE's)p FF(,)f(J.)f(Nonlinear)f(Sci.)h Fa(11)270
4815 y FF(\(2001\),)d(no.)g(1,)g(69{87.)100 4915 y([BaB])40
b(D.)25 b(Bam)n(busi,)g(M.)g(Berti,)f FD(A)j(Birkho)l(of-L)l(ewis)j
(typ)l(e)d(the)l(or)l(em)g(for)g(some)h(Hamiltonian)f(PDE's)p
FF(,)f(to)f(app)r(ear)270 5015 y(on)j(SIAM)g(Journal)e(on)h
(Mathematical)g(Analysis.)100 5114 y([BR])41 b(V.)c(Benci,)h(P)-7
b(.)37 b(Rabino)n(witz,)h FD(Critic)l(al)h(p)l(oint)f(the)l(or)l(ems)g
(for)h(inde\014nite)f(functionals)p FF(,)h(In)n(v)n(en)n(t.)d(Math.)h
(52)270 5214 y(\(1979\),)27 b(no.)g(3,)g(241{273.)100
5313 y([BBi])41 b(M.)26 b(Berti,)h(L.)f(Biasco,)f FD(Perio)l(dic)31
b(solutions)e(of)g(nonline)l(ar)g(wave)g(e)l(quations)g(with)g
(non-monotone)f(for)l(cing)270 5413 y(terms)p FF(,)g(to)f(app)r(ear)g
(on)g(Rend.)h(Mat.)g(Acc.)g(Naz.)f(Lincei.)p eop end
%%Page: 25 25
TeXDict begin 25 24 bop 721 66 a Fr(QUASI-PERIODIC)30
b(SOLUTIONS)i(OF)h(F)n(OR)n(CED)g(W)-9 b(A)g(VE)33 b(EQUA)-6
b(TIONS)543 b(25)100 266 y FF([BBi1])40 b(M.)c(Berti,)g(L.)g(Biasco,)f
FD(F)-6 b(or)l(c)l(e)l(d)37 b(vibr)l(ations)h(of)f(wave)h(e)l(quations)
f(with)g(non-monotone)f(nonline)l(arities)p FF(,)270
365 y(Preprin)n(t)27 b(Sissa)g(2004.)100 465 y([BBV])41
b(M.)29 b(Berti,)g(L.)f(Biasco,)g(E.)g(V)-7 b(aldino)r(ci,)29
b FD(Perio)l(dic)k(orbits)e(close)h(to)f(el)t(liptic)h(tori)f(and)h
(applic)l(ations)h(to)d(the)270 565 y(thr)l(e)l(e-b)l(o)l(dy)g(pr)l
(oblem)p FF(,)f(Annali)f(Scuola)f(Normale)g(Sup.)h(di)g(Pisa,)e(Cl.)i
(Sci.)g(\(V\))g(3,)g(87-138,)d(2004.)100 664 y([BB1])40
b(M.)30 b(Berti,)f(P)-7 b(.)29 b(Bolle,)g FD(Perio)l(dic)k(solutions)e
(of)h(nonline)l(ar)g(wave)g(e)l(quations)f(with)h(gener)l(al)f(nonline)
l(arities)p FF(,)270 764 y(Comm.)d(Math.)g(Ph)n(ys.)e
Fa(243)h FF(\(2003\),)f(no.)i(2,)f(315{328.)100 863 y([BB2])40
b(M.)c(Berti,)i(P)-7 b(.)36 b(Bolle,)h FD(Multiplicity)i(of)f(p)l(erio)
l(dic)i(solutions)d(of)h(nonline)l(ar)g(wave)g(e)l(quations)p
FF(,)h(Nonlinear)270 963 y(Anal.,)28 b(no.)f(56)g(\(2004\),)f
(1011-1046.)100 1063 y([BB3])40 b(M.)d(Berti,)i(P)-7
b(.)37 b(Bolle,)h FD(Cantor)h(families)h(of)f(p)l(erio)l(dic)h
(solutions)f(for)g(c)l(ompletely)g(r)l(esonant)f(non)g(line)l(ar)270
1162 y(wave)31 b(e)l(quations)p FF(,)d(preprin)n(t)f(SISSA)h(\(2004\).)
100 1262 y([BPr])40 b(M.)c(Berti,)h(M.)f(Pro)r(cesi,)h
FD(Quasi-p)l(erio)l(dic)i(oscil)t(lations)g(for)f(wave)g(e)l(quations)g
(with)f(a)h(p)l(erio)l(dic)h(for)l(cing)p FF(,)270 1362
y(preprin)n(t)27 b(2004.)100 1461 y([B1])41 b(J.)e(Bourgain,)i
FD(Quasi-p)l(erio)l(dic)i(solutions)e(of)g(Hamiltonian)h(p)l(erturb)l
(ations)f(of)h(2D)e(line)l(ar)i(Schr\177)-42 b(odinger)270
1561 y(e)l(quations)p FF(,)28 b(Ann.)g(of)g(Math.)g(\(2\))g
Fa(148)e FF(\(1998\),)h(no.)g(2,)g(363{439.)100 1660
y([B2])41 b(J.)30 b(Bourgain,)g FD(Perio)l(dic)35 b(solutions)d(of)i
(nonline)l(ar)f(wave)g(e)l(quations)p FF(,)f(Chicago)e(Lectures)g(in)h
(Math.,)g(Univ.)270 1760 y(Chicago)c(Press,)f(Chicago,)g(IL,)i(1999.)
100 1860 y([BN])41 b(H.)35 b(Brezis,)f(L.)g(Niren)n(b)r(erg,)h
FD(F)-6 b(or)l(c)l(e)l(d)36 b(vibr)l(ations)h(for)f(a)g(nonline)l(ar)g
(wave)h(e)l(quation)p FF(,)f(Comm.)e(Pure)f(Appl.)270
1959 y(Math.)28 b(31)f(\(1978\),)f(no.)h(1,)h(1-30.)100
2059 y([C])64 b(J.)20 b(M.)h(Coron,)f FD(Perio)l(dic)26
b(solutions)d(of)h(a)f(nonline)l(ar)h(wave)g(e)l(quation)f(without)g
(assumption)h(of)f(monotonicity)p FF(,)270 2159 y(Math.)28
b(Ann.)g(262)f(\(1983\),)f(no.)h(2,)g(273{285.)100 2258
y([CW])42 b(W.)e(Craig,)h(E.)f(W)-7 b(a)n(yne,)42 b FD(Newton)-8
b('s)40 b(metho)l(d)i(and)f(p)l(erio)l(dic)i(solutions)e(of)h(nonline)l
(ar)f(wave)h(e)l(quations)p FF(,)270 2358 y(Comm.)28
b(Pure)f(Appl.)h(Math.)g Fa(46)f FF(\(1993\),)f(1409{1498.)100
2457 y([GMP])41 b(G.)29 b(Gen)n(tile,)g(V.)g(Mastropietro,)e(M.)i(Pro)r
(cesi,)e FD(Perio)l(dic)33 b(solutions)d(of)i(c)l(ompletely)g(r)l
(esonant)e(nonline)l(ar)270 2557 y(wave)h(e)l(quations)p
FF(,)d(to)f(app)r(ear)g(in)h(Comm.)f(Math.)h(Ph)n(ys.)100
2657 y([GP])41 b(G.)d(Gen)n(tile,)j(M.)d(Pro)r(cesi,)h
FD(Conservation)i(of)f(r)l(esonant)f(p)l(erio)l(dic)i(solutions)e(for)h
(the)g(one-dimensional)270 2756 y(nonline)l(ar)31 b(Schr\177)-42
b(odinger)31 b(e)l(quation)f FF(,)e(preprin)n(t.)100
2856 y([Mc])41 b(P)-7 b(.J.)33 b(McKenna,)i FD(On)f(solutions)i(of)g(a)
f(nonline)l(ar)h(wave)g(question)f(when)h(the)f(r)l(atio)h(of)g(the)g
(p)l(erio)l(d)g(to)f(the)270 2956 y(length)30 b(of)h(the)f(interval)g
(is)g(irr)l(ational)p FF(,)g(Pro)r(c.)c(Amer.)i(Math.)f(So)r(c.)h(93)f
(\(1985\),)f(no.)h(1,)h(59{64.)100 3055 y([K])59 b(S.B.)41
b(Kuksin,)j FD(A)n(nalysis)f(of)g(Hamiltonian)g(PDEs)p
FF(,)i(Oxford)40 b(Lecture)g(Series)h(in)g(Mathematics)g(and)f(its)270
3155 y(Applications,)28 b(19.)f(Oxford)f(Univ)n(ersit)n(y)h(Press,)f
(2000.)100 3254 y([LS])41 b(B.V.)29 b(Lidski)-9 b(\025)-32
b(\020,)27 b(E.I.)h(Sh)n(ul)1096 3224 y Fi(0)1119 3254
y FF(man,)g FD(Perio)l(dic)33 b(solutions)d(of)h(the)g(e)l(quation)f
Fl(u)2621 3266 y Fh(tt)2694 3254 y Fk(\000)18 b Fl(u)2825
3266 y Fh(xx)2923 3254 y FF(+)g Fl(u)3054 3224 y FB(3)3115
3254 y FF(=)24 b(0,)k(F)-7 b(unct.)29 b(Anal.)270 3354
y(Appl.)g Fa(22)e FF(\(1988\),)f(no.)h(4,)h(332{333.)100
3454 y([PY])41 b(P)-7 b(.I.)28 b(Plotinik)n(o)n(v,)f(L.N.)i(Y)-7
b(ungermann,)28 b FD(Perio)l(dic)33 b(solutions)d(of)h(a)g(we)l(akly)h
(nonline)l(ar)f(wave)g(e)l(quation)g(with)270 3553 y(an)26
b(irr)l(ational)g(r)l(elation)h(of)f(p)l(erio)l(d)h(to)e(interval)h
(length)p FF(,)e(transl.)e(in)h(Di\013.)h(Eq.)f(24)f(\(1988\))f(n.)i(9)
g(pp.)g(1059-1065.)100 3653 y([P])67 b(J.)34 b(P\177)-42
b(osc)n(hel,)34 b FD(Quasi-p)l(erio)l(dic)j(solutions)f(for)g(a)g
(nonline)l(ar)g(wave)g(e)l(quation)p FF(,)g(Commen)n(t.)e(Math.)g
(Helv.)g Fa(71)270 3753 y FF(\(1996\),)27 b(no.)g(2,)g(269{296.)100
3852 y([P1])40 b(M.)33 b(Pro)r(cesi,)f FD(Quasi-p)l(erio)l(dic)k
(solutions)e(for)h(c)l(ompletely)g(r)l(esonant)f(wave)h(e)l(quations)f
(in)h(1D)f(and)g(2D)p FF(,)f(to)270 3952 y(app)r(ear)27
b(in)h(Discr.)f(Con)n(t.)g(Dyn.)i(Syst.)100 4051 y([P2])40
b(M.)19 b(Pro)r(cesi,)g FD(F)-6 b(amilies)23 b(of)g(quasi-p)l(erio)l
(dic)h(solutions)e(for)g(a)g(c)l(ompletely)h(r)l(esonant)e(wave)i(e)l
(quation,)e FF(preprin)n(t.)100 4151 y([R1])41 b(P)-7
b(.)31 b(Rabino)n(witz,)i FD(Perio)l(dic)j(solutions)e(of)g(nonline)l
(ar)g(hyp)l(erb)l(olic)i(p)l(artial)f(di\013er)l(ential)g(e)l(quations)
p FF(,)f(Comm.)270 4251 y(Pure)27 b(Appl.)h(Math.,)g(20)f(1967)f
(145{205.)100 4350 y([R2])41 b(P)-7 b(.)20 b(Rabino)n(witz,)h
FD(Time)i(p)l(erio)l(dic)i(solutions)e(of)h(nonline)l(ar)f(wave)h(e)l
(quations)p FF(,)d(Man)n(uscripta)e(Math.)h(5)g(\(1971\),)270
4450 y(165-194.)100 4550 y([R3])41 b(P)-7 b(.)22 b(H.)g(Rabino)n(witz,)
g FD(Minimax)k(metho)l(ds)f(in)g(critic)l(al)g(p)l(oint)g(the)l(ory)g
(with)h(applic)l(ations)g(to)f(di\013er)l(ential)h(e)l(qua-)270
4649 y(tions)p FF(,)34 b(CBMS)f(Regional)e(Conference)h(Series)f(in)i
(Mathematics,)g(65.)f(American)g(Mathematical)g(So)r(ciet)n(y)-7
b(,)270 4749 y(1986.)100 4848 y([W)g(a])41 b(C.E.)23
b(W)-7 b(a)n(yne,)23 b FD(Perio)l(dic)28 b(and)e(quasi-p)l(erio)l(dic)i
(solutions)e(of)g(nonline)l(ar)g(wave)h(e)l(quations)f(via)g(KAM)g(the)
l(ory)p FF(,)270 4948 y(Comm.)i(Math.)g(Ph)n(ys.)e Fa(127)h
FF(\(1990\),)f(no.)i(3,)f(479{528.)p eop end
%%Trailer

userdict /end-hook known{end-hook}if
%%EOF